patterns in prime numbers: fixed digits repetitionsrgmia.org/papers/v20/v20a17.pdf ·...

Inder J. Taneja RGMIA Research Report Collection, 20(2017), pp.1-75, http://rgmia.org/v20.php

Patterns in Prime Numbers:Fixed Digits Repetitions

Inder J. Taneja1

Abstract

In this paper, we shall work with prime numbers in such a way that they can be extended byinserting digits in repeated ways, and still it continues prime numbers. These types of prime numberscalled ”Patterns in Prime Numbers”. The repetition of digits are with one, two, three, four, five and sixdigits. This work is from 10 to 7 string length 7, i.e., in each case we have 10 to 7 primes numbers withrepeated digits.

The work of this paper is divided in sections and subsections given below:

1 Introduction;1.1 Multiple Prime Patterns;2 Patterns with Strings of Length 10;2.1 Single Digit Repetitions;2.2 Double Digits Repetitions;2.3 Triple Digits Repetitions;2.4 Four Digits Repetitions;3 Patterns with Strings of Length 9;3.1 Single Digit Repetitions;3.2 Two Digits Repetitions;3.3 Three Digits Repetitions;3.4 Four Digits Repetitions;4 Patterns with Strings of Length 8;4.1 Single Digit Repetitions;4.2 Two Digits Repetitions;4.3 Three Digits Repetitions;4.4 Four Digits Repetitions;4.5 Five Digits Repetitions;5 Patterns with Strings of Length 7;5.1 Single Digit Repetitions;5.2 Double Digits Repetitions;5.3 Triple Digits Repetitions;5.4 Four Digits Repetitions;5.5 Five Digits Repetitions;5.6 Six Digits Repetitions.

1Formerly, Professor of Mathematics, Universidade Federal de Santa Catarina, 88.040-900 Florianopolis, SC, Brazil.E-mail: [email protected]; Web-site: www.numbersmagic.wordpress.com.

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RGMIA Res. Rep. Coll. 20 (2017), Art. 17. 75 pp.


1 IntroductionFirst let us understand what we mean by ”patterns in prime numbers”. It is understood as repetitionof same digits in limited continuous way the the resulting number still remains as prime numbers. Seesome examples,

• 71701700170001700001.

• 11116141116141614111614161416141116141614161416141.

• 106910186910181869101818186910181818186910181818181869.

• 474135741351357413513513574135135135135741351351351351357.

Let us analyse the above prime numbers. In the first case digit 0 repeats 4 times. In the second casethe the digit 1614 repeats 4 times. In the third and forth examples, the numbers 18 and 135 repeats 5times respectively. Also to note that in each case the next number in the string are not prime numbers.See below:

700001 = 197× 35533;1161416141614161416141 = 82549× 14069415033666809;

1018181818181869 = 11× 92561983471079;41351351351351351357 = 29× 9461× 597643× 252181271.

Let us examine two interesting examples (refer [2]),

• 31331333133331333331333333133333331.

• 5959959999599999599999995999999999959999999999999.

The difference between two examples is that the first is uniform in increasing single digit 3 to next.In the first case the string can start at any place. The next example in this case is not a prime,

333333331 = 17× 19607843.

The second example is not uniform. In this case the next of similar time prime number is

59999999999999999999999.

2


Our study is limited to the examples of first kind, i.e., uniform patterns. This is divided in sectionsand subsections according to number of primes in each category. This work is limited to 7 to 10 stringsof primes, but is written below in reverse order i.e., from 10 to 7. Thus in each case, single, double,triple, ... digits repetitions are done separately. In each case few examples are given. By no means,these only examples, there might be much more.

1.1 Multiple Prime PatternsBelow is a example of a prime number 50487611, where the pattern of lenght 9 is formed by threedifferent ways, by digits 489, 894 and 948 respectively.

I 50 48761150 489 48761150 489 489 48761150 489 489 489 48761150 489 489 489 489 48761150 489 489 489 489 489 48761150 489 489 489 489 489 489 48761150 489 489 489 489 489 489 489 48761150 489 489 489 489 489 489 489 489 487611.

I 504 87611504 894 87611504 894 894 87611504 894 894 894 87611504 894 894 894 894 87611504 894 894 894 894 894 87611504 894 894 894 894 894 894 87611504 894 894 894 894 894 894 894 87611504 894 894 894 894 894 894 894 894 87611.

I 5048 76115048 948 76115048 948 948 76115048 948 948 948 76115048 948 948 948 948 76115048 948 948 948 948 948 76115048 948 948 948 948 948 948 76115048 948 948 948 948 948 948 948 76115048 948 948 948 948 948 948 948 948 7611.

We observe that in all the 3 cases the primes are the same in each line, for example, the last one ineach case is 50489489489489489489489489487611. Only difference is in way the digits for repetitionare choosed. This is due the fact that 48 is appearing in the prime as well as in the repetition digit:5048(948)7611. For simplicity, let us call the as ”multiple pattern prime numbers”. The detailed studyon this type of prime numbers shall be dealt elsewhere.

3


For study on numbers in different situations can be seen in author’s work [6]. This work is summary ofauthor’s on numbers work for the past few years. See also author’s recent work [7] on Hardy-Ramanujannumber 1729. This work brings the number 1729 in different situations. From historical point of view,refer [1, 2, 4, 5].

2 Patterns with Strings of Length 10In this section, we shall give examples of patterns with different digits repetitions for primes length 10.This has be divided according digits in the subsections below.

2.1 Single Digit RepetitionsIn this case, we have only one example,

I 291234 73291234 3 73291234 3 3 73291234 3 3 3 73291234 3 3 3 3 73291234 3 3 3 3 3 73291234 3 3 3 3 3 3 73291234 3 3 3 3 3 3 3 73291234 3 3 3 3 3 3 3 3 73291234 3 3 3 3 3 3 3 3 3 73.

There might be more examples, but we found only one.

2.2 Two Digits RepetitionsIn this case also we have only one example,

I 125931 97125931 33 97125931 33 33 97125931 33 33 33 97125931 33 33 33 33 97125931 33 33 33 33 33 97125931 33 33 33 33 33 33 97125931 33 33 33 33 33 33 33 97125931 33 33 33 33 33 33 33 33 97125931 33 33 33 33 33 33 33 33 33 97.

Again, there might be more examples of this kind.

2.3 Three Digits RepetitionsBelow are few examples of 3 digits repetitions.

4


I 7 549937 678 549937 678 678 549937 678 678 678 549937 678 678 678 678 549937 678 678 678 678 678 549937 678 678 678 678 678 678 549937 678 678 678 678 678 678 678 549937 678 678 678 678 678 678 678 678 549937 678 678 678 678 678 678 678 678 678 54993.

I 1 4077271 105 4077271 105 105 4077271 105 105 105 4077271 105 105 105 105 4077271 105 105 105 105 105 4077271 105 105 105 105 105 105 4077271 105 105 105 105 105 105 105 4077271 105 105 105 105 105 105 105 105 4077271 105 105 105 105 105 105 105 105 105 407727.

I 56 2993356 218 2993356 218 218 2993356 218 218 218 2993356 218 218 218 218 2993356 218 218 218 218 218 2993356 218 218 218 218 218 218 2993356 218 218 218 218 218 218 218 2993356 218 218 218 218 218 218 218 218 2993356 218 218 218 218 218 218 218 218 218 29933.

I 562 9933562 192 9933562 192 192 9933562 192 192 192 9933562 192 192 192 192 9933562 192 192 192 192 192 9933562 192 192 192 192 192 192 9933562 192 192 192 192 192 192 192 9933562 192 192 192 192 192 192 192 192 9933562 192 192 192 192 192 192 192 192 192 9933.

I 65 9646165 924 9646165 924 924 9646165 924 924 924 9646165 924 924 924 924 9646165 924 924 924 924 924 9646165 924 924 924 924 924 924 9646165 924 924 924 924 924 924 924 9646165 924 924 924 924 924 924 924 924 9646165 924 924 924 924 924 924 924 924 924 96461.

I 659 6461659 249 6461659 249 249 6461659 249 249 249 6461659 249 249 249 249 6461659 249 249 249 249 249 6461659 249 249 249 249 249 249 6461659 249 249 249 249 249 249 249 6461659 249 249 249 249 249 249 249 249 6461659 249 249 249 249 249 249 249 249 249 6461.

I 120623 77120623 945 77120623 945 945 77120623 945 945 945 77120623 945 945 945 945 77120623 945 945 945 945 945 77120623 945 945 945 945 945 945 77120623 945 945 945 945 945 945 945 77120623 945 945 945 945 945 945 945 945 77120623 945 945 945 945 945 945 945 945 945 77.

I 1403284 71403284 135 71403284 135 135 71403284 135 135 135 71403284 135 135 135 135 71403284 135 135 135 135 135 71403284 135 135 135 135 135 135 71403284 135 135 135 135 135 135 135 71403284 135 135 135 135 135 135 135 135 71403284 135 135 135 135 135 135 135 135 135 7.

5


I 16608 56916608 240 56916608 240 240 56916608 240 240 240 56916608 240 240 240 240 56916608 240 240 240 240 240 56916608 240 240 240 240 240 240 56916608 240 240 240 240 240 240 240 56916608 240 240 240 240 240 240 240 240 56916608 240 240 240 240 240 240 240 240 240 569.

I 20809493255 20809493255 255 20809493255 255 255 20809493255 255 255 255 20809493255 255 255 255 255 20809493255 255 255 255 255 255 20809493255 255 255 255 255 255 255 20809493255 255 255 255 255 255 255 255 20809493255 255 255 255 255 255 255 255 255 20809493.

I 2 08094932 552 08094932 552 552 08094932 552 552 552 08094932 552 552 552 552 08094932 552 552 552 552 552 08094932 552 552 552 552 552 552 08094932 552 552 552 552 552 552 552 08094932 552 552 552 552 552 552 552 552 08094932 552 552 552 552 552 552 552 552 552 0809493.

I 3100 33213100 843 33213100 843 843 33213100 843 843 843 33213100 843 843 843 843 33213100 843 843 843 843 843 33213100 843 843 843 843 843 843 33213100 843 843 843 843 843 843 843 33213100 843 843 843 843 843 843 843 843 33213100 843 843 843 843 843 843 843 843 843 3321.

I 34284 52134284 699 52134284 699 699 52134284 699 699 699 52134284 699 699 699 699 52134284 699 699 699 699 699 52134284 699 699 699 699 699 699 52134284 699 699 699 699 699 699 699 52134284 699 699 699 699 699 699 699 699 52134284 699 699 699 699 699 699 699 699 699 521.

I 352 96867352 357 96867352 357 357 96867352 357 357 357 96867352 357 357 357 357 96867352 357 357 357 357 357 96867352 357 357 357 357 357 357 96867352 357 357 357 357 357 357 357 96867352 357 357 357 357 357 357 357 357 96867352 357 357 357 357 357 357 357 357 357 96867.

I 4199233 34199233 279 34199233 279 279 34199233 279 279 279 34199233 279 279 279 279 34199233 279 279 279 279 279 34199233 279 279 279 279 279 279 34199233 279 279 279 279 279 279 279 34199233 279 279 279 279 279 279 279 279 34199233 279 279 279 279 279 279 279 279 279 3.

I 46089 97746089 774 97746089 774 774 97746089 774 774 774 97746089 774 774 774 774 97746089 774 774 774 774 774 97746089 774 774 774 774 774 774 97746089 774 774 774 774 774 774 774 97746089 774 774 774 774 774 774 774 774 97746089 774 774 774 774 774 774 774 774 774 977.

6


I 59 29000959 273 29000959 273 273 29000959 273 273 273 29000959 273 273 273 273 29000959 273 273 273 273 273 29000959 273 273 273 273 273 273 29000959 273 273 273 273 273 273 273 29000959 273 273 273 273 273 273 273 273 29000959 273 273 273 273 273 273 273 273 273 290009.

I 592 90009592 732 90009592 732 732 90009592 732 732 732 90009592 732 732 732 732 90009592 732 732 732 732 732 90009592 732 732 732 732 732 732 90009592 732 732 732 732 732 732 732 90009592 732 732 732 732 732 732 732 732 90009592 732 732 732 732 732 732 732 732 732 90009.

I 630842 27630842 558 27630842 558 558 27630842 558 558 558 27630842 558 558 558 558 27630842 558 558 558 558 558 27630842 558 558 558 558 558 558 27630842 558 558 558 558 558 558 558 27630842 558 558 558 558 558 558 558 558 27630842 558 558 558 558 558 558 558 558 558 27.

I 6 82498916 978 82498916 978 978 82498916 978 978 978 82498916 978 978 978 978 82498916 978 978 978 978 978 82498916 978 978 978 978 978 978 82498916 978 978 978 978 978 978 978 82498916 978 978 978 978 978 978 978 978 82498916 978 978 978 978 978 978 978 978 978 8249891.

I 6991474 96991474 225 96991474 225 225 96991474 225 225 225 96991474 225 225 225 225 96991474 225 225 225 225 225 96991474 225 225 225 225 225 225 96991474 225 225 225 225 225 225 225 96991474 225 225 225 225 225 225 225 225 96991474 225 225 225 225 225 225 225 225 225 9.

I 700 39153700 738 39153700 738 738 39153700 738 738 738 39153700 738 738 738 738 39153700 738 738 738 738 738 39153700 738 738 738 738 738 738 39153700 738 738 738 738 738 738 738 39153700 738 738 738 738 738 738 738 738 39153700 738 738 738 738 738 738 738 738 738 39153.

I 765 48337765 684 48337765 684 684 48337765 684 684 684 48337765 684 684 684 684 48337765 684 684 684 684 684 48337765 684 684 684 684 684 684 48337765 684 684 684 684 684 684 684 48337765 684 684 684 684 684 684 684 684 48337765 684 684 684 684 684 684 684 684 684 48337.

I 827115 23827115 702 23827115 702 702 23827115 702 702 702 23827115 702 702 702 702 23827115 702 702 702 702 702 23827115 702 702 702 702 702 702 23827115 702 702 702 702 702 702 702 23827115 702 702 702 702 702 702 702 702 23827115 702 702 702 702 702 702 702 702 702 23.

7


I 9 77880019 831 77880019 831 831 77880019 831 831 831 77880019 831 831 831 831 77880019 831 831 831 831 831 77880019 831 831 831 831 831 831 77880019 831 831 831 831 831 831 831 77880019 831 831 831 831 831 831 831 831 77880019 831 831 831 831 831 831 831 831 831 7788001.

2.4 Four Digits RepetitionsIn this case, we have only one prime pattern with 4 digits repetitions.

I 1179 18831179 7902 18831179 7902 7902 18831179 7902 7902 7902 18831179 7902 7902 7902 7902 18831179 7902 7902 7902 7902 7902 18831179 7902 7902 7902 7902 7902 7902 18831179 7902 7902 7902 7902 7902 7902 7902 18831179 7902 7902 7902 7902 7902 7902 7902 7902 18831179 7902 7902 7902 7902 7902 7902 7902 7902 7902 1883.

There may be much more of this kind, but we found only one.


3.1 Single Digit RepetitionsBelow are few examples, with simplified notations.

8


• [6073, 69073; 9] I 6 0736 9 0736 9 9 0736 9 9 9 0736 9 9 9 9 0736 9 9 9 9 9 0736 9 9 9 9 9 9 0736 9 9 9 9 9 9 9 0736 9 9 9 9 9 9 9 9 073.

• [47881, 437881; 9] I 4 78814 3 78814 3 3 78814 3 3 3 78814 3 3 3 3 78814 3 3 3 3 3 78814 3 3 3 3 3 3 78814 3 3 3 3 3 3 3 78814 3 3 3 3 3 3 3 3 7881.

Above two patterns we have written in two different ways. Left side is simplified notation, whileright side is complete patterns. Instead writing whole the pattern, we can write is int reduced formrepresenting the same way. The left side is understood as

[A, B; k ]where A is prime numbers, B is prime number with repetition of digits. The letter k represents the totalprime numbers. In the above two cases, k=9 . Thus, above two examples can equivalently be written as,

• [6073, 69073; 9]• [47881, 437881; 9].

From now onwards, we shall use frequently, these noations. Below are more examples,

• [1262237, 16262237; 9].• [1838591, 18638591; 9].• [3569977, 35069977; 9].• [4128253, 94128253; 9].

• [19821041, 198210431; 9].• [21728671, 217028671; 9].• [24520651, 245620651; 9].

• [25320433, 253230433; 9].• [29478133, 294781633; 9].• [31199611, 311990611; 9].• [31251853, 312561853; 9].• [39911437, 399114337; 9].• [39911437, 399114337; 9].• [42280673, 422980673; 9].• [46333559, 463335569; 9].• [48344561, 438344561; 9].

• [50490133, 504900133; 9].• [50490133, 504900133; 9].• [54252893, 542523893; 9].• [55376401, 559376401; 9].• [58359341, 586359341; 9].• [60822539, 608262539; 9].• [62929117, 629329117; 9].• [70249343, 702349343; 9].• [94369151, 943691531; 9].

3.2 Two Digits RepetitionsBelow are examples prime patterns with 2 digits repetitions :

• [484067, 48403967; 9] I 4840 674840 39 674840 39 39 674840 39 39 39 674840 39 39 39 39 674840 39 39 39 39 39 674840 39 39 39 39 39 39 674840 39 39 39 39 39 39 39 674840 39 39 39 39 39 39 39 39 67.

9


• [2521747, 259921747; 9] I 25 2174725 99 2174725 99 99 2174725 99 99 99 2174725 99 99 99 99 2174725 99 99 99 99 99 2174725 99 99 99 99 99 99 2174725 99 99 99 99 99 99 99 2174725 99 99 99 99 99 99 99 99 21747.

Below are more examples in reduced form:

• [5671811, 567571811; 9].• [5671811, 567571811; 9].• [8452481, 845482481; 9].• [8972069, 897206489; 9].

• [13161593, 1316199593; 9].• [15246053, 1524756053; 9].• [18794393, 4518794393; 9].• [20429033, 2042903573; 9].• [28054067, 2805406187; 9].• [44780807, 4724780807; 9].

• [44972273, 4497226373; 9].• [45342449, 4534244249; 9].• [45342449, 4534244249; 9].• [45520051, 4552005163; 9].• [45711593, 4581711593; 9].• [47974319, 3347974319; 9].• [58254913, 5822154913; 9].• [66616939, 6624616939; 9].• [67179227, 6760179227; 9].• [71274997, 1271274997; 9].

• [72407837, 7251407837; 9].• [85917631, 8365917631; 9].• [87293977, 8778293977; 9].• [88884787, 8888479987; 9].• [90715591, 9076615591; 9].• [92553907, 9255330907; 9].• [95675141, 9215675141; 9].• [96379757, 9637459757; 9].• [96947023, 9694709923; 9].• [99387037, 9759387037; 9].

3.3 Three Digits RepetitionsBelow are examples prime patterns with three digits repetitions :

• [1907, 8671907; 9] I 1907867 1907867 867 1907867 867 867 1907867 867 867 867 1907867 867 867 867 867 1907867 867 867 867 867 867 1907867 867 867 867 867 867 867 1907867 867 867 867 867 867 867 867 1907.

10


• [137623, 747137623; 9] I 137623747 137623747 747 137623747 747 747 137623747 747 747 747 137623747 747 747 747 747 137623747 747 747 747 747 747 137623747 747 747 747 747 747 747 137623747 747 747 747 747 747 747 747 137623.


• [288467, 288372467; 9].• [290593, 290723593; 9].• [425653, 425651383; 9].• [493231, 498523231; 9].• [604249, 674704249; 9].• [662551, 662551843; 9].

• [711131, 711987131; 9].• [879247, 825879247; 9].• [879247, 825879247; 9].• [979553, 912679553; 9].• [980963, 980930963; 9].• [980963, 980930963; 9].

• [1047751, 4501047751; 9].• [1249111, 1246429111; 9].• [1337617, 1987337617; 9].• [1658389, 1655558389; 9].• [1658389, 1655558389; 9].

• [2141669, 2141662049; 9].• [2621467, 2621417167; 9].• [2633131, 2507633131; 9].• [2822011, 2822019241; 9].• [3145487, 3145654487; 9].

• [3253469, 3253449569; 9].• [3328541, 3328547591; 9].• [4019689, 4019658289; 9].• [4538627, 4538621837; 9].• [4654459, 4390654459; 9].

• [4782061, 4782066451; 9].• [5706853, 5706855463; 9].• [5824417, 5824996417; 9].• [5913797, 5913796907; 9].• [5948909, 5948923109; 9].

• [6103057, 6103056097; 9].• [6173369, 6264173369; 9].• [6673993, 6673105993; 9].• [6695947, 6694745947; 9].

• [7169131, 7169483131; 9].• [7449011, 7483749011; 9].• [7830041, 7830702041; 9].• [8075849, 8075846609; 9].

• [8360377, 8362160377; 9].• [8360377, 8362160377; 9].• [8652187, 8652182257; 9].• [8672333, 8632772333; 9].

• [8942963, 8942195963; 9].• [9113147, 9118163147; 9].• [9563507, 8049563507; 9].• [9763693, 9763231693; 9].

• [10160861, 10160390861; 9].• [10219679, 10219672439; 9].• [10272991, 10272696991; 9].• [10425313, 10425555313; 9].• [10425313, 10425555313; 9].

• [10620991, 10620354991; 9].• [10984559, 80110984559; 9].• [10988797, 10183988797; 9].• [11049641, 11049234641; 9].• [11836397, 18371836397; 9].

• [11857267, 11857984267; 9].• [12082663, 18432082663; 9].• [12590881, 12590887471; 9].• [12822181, 12853422181; 9].• [13157723, 13156997723; 9].

• [13221337, 13165221337; 9].• [13482041, 46813482041; 9].• [13827277, 13762827277; 9].• [14068081, 17804068081; 9].• [14069849, 14069844989; 9].

• [14094413, 14062194413; 9].• [14740211, 14740211561; 9].• [14740211, 14740211561; 9].• [15091231, 15099451231; 9].• [15141283, 15141516283; 9].

• [15423103, 13665423103; 9].• [16563431, 16563437201; 9].• [17159201, 12437159201; 9].• [17165231, 17372165231; 9].• [17434019, 13247434019; 9].

11


• [17612953, 17612917753; 9].• [17701583, 17483701583; 9].• [17852767, 17537852767; 9].• [17852767, 17537852767; 9].• [18194431, 18153494431; 9].

• [18555487, 18555489907; 9].• [18847193, 41118847193; 9].• [19175927, 19124375927; 9].• [19307681, 19305587681; 9].• [19704011, 19745304011; 9].

• [20142079, 20126442079; 9].• [20158961, 20158982861; 9].• [20492041, 23990492041; 9].• [20868079, 20868078139; 9].• [20988571, 28910988571; 9].

• [21004603, 21246004603; 9].′

• [21072133, 24891072133; 9].• [21087173, 28641087173; 9].• [21468289, 21468282529; 9].• [21768749, 21769998749; 9].

• [21877561, 21877566661; 9].• [21877561, 21877566661; 9].• [22541669, 23212541669; 9].• [23106197, 72323106197; 9].• [23149267, 23149675267; 9].

• [23401853, 22973401853; 9].• [24388849, 24388849393; 9].• [25203989, 25203989273; 9].• [25947773, 25168947773; 9].• [26458259, 39626458259; 9].

• [26740667, 26471740667; 9].• [26756701, 26742356701; 9].• [27037781, 27037782491; 9].• [27194171, 27194192471; 9].• [27235423, 27123235423; 9].

• [27535367, 27535367507; 9].• [27535367, 27535367507; 9].• [28070929, 28070997529; 9].• [28157509, 26938157509; 9].• [29944349, 29944618349; 9].

• [30631619, 30631216619; 9].• [30999341, 30999342641; 9].• [30999341, 30999342641; 9].• [31215829, 67231215829; 9].• [31224539, 31224562139; 9].

• [31330391, 31330392461; 9].• [32110097, 32110030397; 9].• [32199589, 32161899589; 9].• [32689199, 62732689199; 9].• [32952433, 22532952433; 9].

• [34062481, 34062528481; 9].• [34203821, 34203821357; 9].• [35505761, 35504745761; 9].• [35871169, 35871150769; 9].• [36831203, 36831241103; 9].

• [37028423, 37028423261; 9].• [37068631, 38257068631; 9].• [38177423, 38177426033; 9].• [38470097, 39518470097; 9].• [38843027, 38842823027; 9].

• [39803821, 21039803821; 9].• [39897841, 39897841747; 9].• [39919241, 39919633241; 9].• [40169609, 40903169609; 9].• [40442819, 68740442819; 9].

• [40670549, 40670852549; 9].• [40702327, 40757002327; 9].• [41404801, 41402494801; 9].• [41723579, 41426723579; 9].• [42138883, 42138886393; 9].

• [42436651, 42436657411; 9].• [42797761, 42354797761; 9].• [43306811, 43306936811; 9].• [43306811, 43306936811; 9].• [43328603, 43753328603; 9].

• [43328603, 43753328603; 9].• [43332307, 43337592307; 9].• [44004413, 44004621413; 9].• [44009003, 44195009003; 9].• [44244769, 44245374769; 9].

• [44797967, 44797962977; 9].• [44810989, 44810954989; 9].• [44810989, 44810954989; 9].• [45347821, 45347813821; 9].• [45347821, 45347813821; 9].

• [45372391, 45372345391; 9].• [45372391, 45372345391; 9].• [45619547, 45619327547; 9].• [45662767, 62145662767; 9].• [46074937, 46507074937; 9].

• [46179439, 26146179439; 9].• [46294229, 46292764229; 9].• [46848259, 46486848259; 9].• [46848259, 46486848259; 9].• [47554097, 47554459097; 9].

• [47627777, 47627273777; 9].• [47773813, 45227773813; 9].• [48927863, 46788927863; 9].• [49150501, 49176850501; 9].• [49313687, 49312373687; 9].

• [49423433, 49423888433; 9].• [49700491, 49717700491; 9].• [49700491, 49717700491; 9].• [50487611, 50489487611; 9].• [50487611, 50489487611; 9].

12


• [50487611, 50489487611; 9].• [51426209, 16851426209; 9].• [53744533, 53741954533; 9].• [53937589, 53937585139; 9].• [53984213, 53984218553; 9].

• [54017989, 54015797989; 9].• [54247367, 54247360667; 9].• [54532237, 54532236127; 9].• [54678049, 54677688049; 9].• [54707503, 52374707503; 9].

• [54810473, 54810477173; 9].• [54810473, 54810477173; 9].• [55255159, 55255158709; 9].• [55537519, 25255537519; 9].• [55992109, 55992129709; 9].

• [57518501, 57539618501; 9].• [57825869, 66657825869; 9].• [57930569, 57918630569; 9].• [57944881, 54627944881; 9].• [58641097, 58641189097; 9].

• [60108383, 60191508383; 9].• [60110213, 67980110213; 9].• [60219487, 65400219487; 9].• [60375839, 66660375839; 9].• [60375839, 66660375839; 9].

• [60652547, 61260652547; 9].• [60652547, 61260652547; 9].• [63364501, 63123364501; 9].• [63364501, 63123364501; 9].• [63390359, 63390366059; 9].

• [63714103, 63714109903; 9].• [63937681, 63937687591; 9].• [64091509, 64795091509; 9].• [64266847, 64266786847; 9].• [64266847, 64266786847; 9].

• [64535101, 64535438101; 9].• [64960463, 64912060463; 9].• [65693657, 65483693657; 9].• [65782471, 65782415371; 9].• [66173141, 66173148581; 9].

• [67666127, 67666127783; 9].• [68275703, 68702275703; 9].• [68376131, 68376174731; 9].• [68463959, 66638463959; 9].• [68633737, 68210633737; 9].

• [68883707, 68588883707; 9].• [68883707, 68588883707; 9].• [69178289, 32169178289; 9].• [70471351, 70479301351; 9].• [71329277, 36371329277; 9].

• [72286901, 79302286901; 9].• [72770767, 72770774467; 9].• [73137851, 73114737851; 9].• [73372441, 73398772441; 9].• [73442491, 73442491669; 9].

• [74851043, 74851588043; 9].• [74929549, 40874929549; 9].• [75344069, 75344061269; 9].• [75344069, 75344061269; 9].• [76385677, 76385675917; 9].

• [76456481, 76469656481; 9].• [76844653, 76847054653; 9].• [77480737, 77480789437; 9].• [77636437, 77633756437; 9].• [77795969, 77794205969; 9].

• [77796541, 87677796541; 9].• [77839303, 77839711303; 9].• [77895481, 79547895481; 9].• [78213353, 78264213353; 9].• [78213353, 78264213353; 9].

• [78917243, 78917245553; 9].• [80674207, 99380674207; 9].• [80745949, 80748705949; 9].• [81478753, 82281478753; 9].• [81478753, 82281478753; 9].

• [81596597, 81596590997; 9].• [82373651, 82373616551; 9].• [82637473, 82637479543; 9].• [83024729, 83028494729; 9].• [83815051, 83815056151; 9].

• [83815051, 83815056151; 9].• [83999821, 83999381821; 9].• [85914359, 85914192359; 9].• [86664541, 86664234541; 9].• [86664541, 86664234541; 9].

• [86993689, 86937893689; 9].• [87084211, 87813084211; 9].• [87282203, 87585282203; 9].• [87813529, 87812313529; 9].• [87813529, 87812313529; 9].

• [88384657, 89878384657; 9].• [89638651, 89638651693; 9].• [90260881, 90972260881; 9].• [90383159, 90183383159; 9].• [90571307, 98520571307; 9].

• [91214033, 91214034893; 9].• [91350653, 91350636053; 9].• [91469593, 94141469593; 9].• [91854941, 29491854941; 9].• [91941043, 34891941043; 9].

• [92307497, 92307495977; 9].• [92513009, 92627513009; 9].• [92592697, 51692592697; 9].• [92832871, 92832847771; 9].• [92874007, 92594874007; 9].

13


• [93458951, 98673458951; 9].• [94252693, 94252692043; 9].• [95170349, 95170361849; 9].• [95177627, 95678177627; 9].• [95324869, 95332724869; 9].• [95485721, 93335485721; 9].

• [95551873, 95586151873; 9].• [96479783, 96479783693; 9].• [96479783, 96479783693; 9].• [96709777, 72696709777; 9].• [97221431, 97221117431; 9].• [97320479, 97320494879; 9].

• [98064181, 98064188071; 9].• [98676173, 98672316173; 9].• [99402829, 11499402829; 9].• [99430469, 99430246469; 9].• [99504103, 18999504103; 9].• [99972287, 99976362287; 9].

3.4 Four Digits RepetitionsBelow are examples prime patterns with four digits repetitions :

• [104803, 1048419403; 9] I 1048 031048 4194 031048 4194 4194 031048 4194 4194 4194 031048 4194 4194 4194 4194 031048 4194 4194 4194 4194 4194 031048 4194 4194 4194 4194 4194 4194 031048 4194 4194 4194 4194 4194 4194 4194 031048 4194 4194 4194 4194 4194 4194 4194 4194 03.

• [329969, 3299632769; 9] I 3299 693299 6327 693299 6327 6327 693299 6327 6327 6327 693299 6327 6327 6327 6327 693299 6327 6327 6327 6327 6327 693299 6327 6327 6327 6327 6327 6327 693299 6327 6327 6327 6327 6327 6327 6327 693299 6327 6327 6327 6327 6327 6327 6327 6327 69.


• [329969, 3299632769; 9].• [583417, 5839834417; 9].

• [1626467, 16264676873; 9].• [1688969, 16849508969; 9].• [2084561, 25085084561; 9].• [2250509, 22505092901; 9].• [2427863, 76232427863; 9].

• [2577409, 25772388409; 9].• [2612921, 26129215727; 9].• [2622863, 26228640863; 9].• [2622863, 26228640863; 9].• [2622863, 26228640863; 9].• [2694287, 26941692287; 9].• [2821451, 60002821451; 9].• [3468533, 34685247533; 9].

• [3468533, 34685247533; 9].• [3684101, 31386684101; 9].• [3716747, 36711716747; 9].• [4191091, 41910923101; 9].• [4710599, 47880810599; 9].• [5017423, 50138527423; 9].• [5307353, 53073344153; 9].

14


• [5846153, 58468322153; 9].• [6364433, 63646831433; 9].• [6815321, 68156120321; 9].• [7095437, 70954373201; 9].• [7728031, 77511228031; 9].

• [8154877, 81545646877; 9].• [8417243, 84132677243; 9].• [9034471, 90354184471; 9].• [9110977, 51639110977; 9].• [9997523, 98694997523; 9].

• [10050197, 100565370197; 9].• [10106557, 101065361857; 9].• [10136839, 101464436839; 9].• [10273343, 219610273343; 9].• [10869451, 154900869451; 9].

• [11299711, 137291299711; 9].• [11594281, 113168594281; 9].• [11798023, 117144998023; 9].• [11839661, 118396612067; 9].• [12392137, 123921387847; 9].

• [12763469, 127634611319; 9].• [14250109, 142501095703; 9].• [14310887, 806714310887; 9].• [14353907, 144671353907; 9].• [14588983, 145889826493; 9].

• [15345569, 153719445569; 9].• [15547193, 346515547193; 9].• [15875527, 283815875527; 9].• [16166707, 161719466707; 9].• [16359059, 163569879059; 9].

• [16422167, 164246202167; 9].• [16501643, 165016438217; 9].• [17183833, 171884903833; 9].• [17775449, 135647775449; 9].• [17958293, 331217958293; 9].

• [18359791, 183597490591; 9].• [19179409, 149509179409; 9].• [20374943, 203749483463; 9].• [20644229, 206442763529; 9].• [20689381, 205511689381; 9].

• [20804591, 208073264591; 9].• [21182807, 211828036337; 9].• [21805547, 218501905547; 9].• [23013589, 236831013589; 9].• [23307379, 233074833379; 9].

• [23654471, 290813654471; 9].• [23959421, 239604859421; 9].• [24589703, 245897353703; 9].• [24589703, 245897353703; 9].• [25441249, 536725441249; 9].

• [25506881, 293575506881; 9].• [25685497, 963625685497; 9].• [26617361, 266739217361; 9].• [27107879, 234987107879; 9].• [28109747, 243538109747; 9].

• [28124923, 283180124923; 9].• [28482071, 284825916071; 9].• [29055017, 290550135857; 9].• [29179681, 291462079681; 9].• [29367049, 293670456619; 9].

• [29712217, 293351712217; 9].• [29757367, 295733757367; 9].• [29872853, 298728513263; 9].• [30803387, 308031167387; 9].• [31603843, 351151603843; 9].• [32170939, 321701200939; 9].

• [32523509, 325210743509; 9].• [33354149, 333573264149; 9].• [34404133, 347353404133; 9].• [35566859, 355616176859; 9].• [36063359, 360625983359; 9].• [37338047, 373380784247; 9].• [37344049, 373441086049; 9].


4.1 Single Digit RepetitionsBelow are few examples, with simplified notation.

15


• [4517, 45317; 8] I 45, 1745, 3 1745, 3 3 1745, 3 3 3 1745, 3 3 3 3 1745, 3 3 3 3 3 1745, 3 3 3 3 3 3 1745, 3 3 3 3 3 3 3 17.

• [7013, 37013; 8] I 70133 70133 3 70133 3 3 70133 3 3 3 70133 3 3 3 3 70133 3 3 3 3 3 70133 3 3 3 3 3 3 7013.

Examples below are according to notations given on left side of the expression:

• [35227, 352327; 8].• [69073, 699073; 8].• [69073, 699073; 8].

• [347707, 6347707; 8].• [437881, 4337881; 8].• [437881, 4337881; 8].• [448807, 4480807; 8].• [615679, 6156739; 8].• [989719, 9089719; 8].

• [1204453, 12604453; 8].• [1231631, 12931631; 8].• [1309067, 13090067; 8].• [1309067, 13090067; 8].• [2062243, 20622463; 8].

• [2294429, 22964429; 8].• [2779771, 27769771; 8].• [3022391, 30223691; 8].• [3557483, 35574893; 8].• [3752191, 37520191; 8].

• [3935027, 39359027; 8].• [4175803, 41758303; 8].• [4340977, 43406977; 8].• [4341497, 43414979; 8].

• [5186219, 51862919; 8].• [5345311, 53435311; 8].• [6503737, 69503737; 8].• [6789037, 67893037; 8].• [6846677, 68436677; 8].

• [7386833, 73868633; 8].• [7661993, 76619903; 8].• [8210987, 38210987; 8].• [8340887, 83460887; 8].• [8648677, 86486707; 8].

• [9276257, 92736257; 8].• [9381259, 93831259; 8].• [9422837, 94228397; 8].• [9776951, 97769513; 8].• [9866449, 98696449; 8].

• [10491709, 130491709; 8].• [10878383, 108798383; 8].• [11157467, 111579467; 8].• [12150431, 121590431; 8].• [12306743, 123067493; 8].

• [12425417, 132425417; 8].• [12442271, 124422713; 8].• [12556613, 125956613; 8].• [12777563, 127773563; 8].• [13153513, 131535133; 8].

• [13153513, 131535133; 8].• [13172983, 163172983; 8].• [13627301, 136273013; 8].• [13742381, 137423831; 8].• [15465071, 154695071; 8].

• [16262237, 166262237; 8].• [16262237, 166262237; 8].• [16715887, 196715887; 8].• [17436709, 170436709; 8].• [17521117, 175291117; 8].

• [17631821, 176391821; 8].• [18342889, 183462889; 8].• [18638591, 186638591; 8].• [18638591, 186638591; 8].• [18753979, 187539799; 8].

• [18753979, 187539799; 8].• [19182203, 191822303; 8].• [19277119, 192737119; 8].• [19381381, 193891381; 8].• [19423373, 194233373; 8].

• [19423373, 194233373; 8].• [19423373, 194233373; 8].• [20215171, 230215171; 8].• [20254049, 202954049; 8].• [20287889, 202807889; 8].

• [20321183, 203621183; 8].• [20511217, 205311217; 8].• [20644319, 209644319; 8].• [20947727, 209477207; 8].• [22183867, 221839867; 8].

16


• [22220777, 222203777; 8].• [22288993, 222889939; 8].• [22328941, 223289431; 8].• [23233349, 623233349; 8].• [23360759, 233607593; 8].

• [23841127, 238411627; 8].• [24619207, 246192097; 8].• [24711163, 247911163; 8].• [24712439, 247012439; 8].• [25743331, 259743331; 8].

• [25889393, 258839393; 8].• [26209181, 260209181; 8].• [26426357, 269426357; 8].• [26611517, 236611517; 8].• [27192013, 271920613; 8].

• [27315641, 273135641; 8].• [28020569, 280020569; 8].• [28020569, 280020569; 8].• [29017721, 239017721; 8].• [29353229, 293653229; 8].

• [29431163, 209431163; 8].• [29815033, 298150933; 8].• [31589407, 310589407; 8].• [31737151, 317376151; 8].• [32262499, 322602499; 8].

• [33197327, 331973627; 8].• [34468013, 340468013; 8].• [34748029, 347480239; 8].• [34752841, 347952841; 8].• [35069977, 350069977; 8].

• [35069977, 350069977; 8].• [35399443, 350399443; 8].• [35703719, 305703719; 8].• [35923967, 359263967; 8].• [36257587, 362957587; 8].

• [36450853, 364050853; 8].• [37688891, 376888913; 8].• [38385251, 383835251; 8].• [39804473, 939804473; 8].• [40579489, 405799489; 8].

• [40579489, 405799489; 8].• [41038133, 410338133; 8].• [41038133, 410338133; 8].• [41592127, 461592127; 8].• [41768341, 461768341; 8].

• [41779471, 417079471; 8].• [42070159, 420070159; 8].• [42070159, 420070159; 8].• [42212459, 422102459; 8].• [42624289, 426243289; 8].

• [42932399, 420932399; 8].• [43813963, 438613963; 8].• [44214413, 443214413; 8].• [44956339, 443956339; 8].• [45556349, 645556349; 8].

• [49740547, 469740547; 8].• [50299177, 502991077; 8].• [50379751, 503769751; 8].• [51099989, 510909989; 8].• [51139637, 511396307; 8].

• [52077401, 520377401; 8].• [52624477, 520624477; 8].• [53032957, 530329507; 8].• [53373797, 536373797; 8].• [53912549, 536912549; 8].

• [54073501, 540736501; 8].• [54535027, 545359027; 8].• [54565271, 545605271; 8].• [54840217, 540840217; 8].• [55121159, 551921159; 8].

• [55826143, 558926143; 8].• [56353057, 563563057; 8].• [56467423, 560467423; 8].• [57014527, 567014527; 8].• [57884633, 957884633; 8].

• [58250707, 582500707; 8].• [58250707, 582500707; 8].• [58675261, 586753261; 8].• [58718213, 587182163; 8].• [58927937, 589027937; 8].

• [59131403, 591316403; 8].• [60314641, 603143641; 8].• [60500371, 605003071; 8].• [60911197, 606911197; 8].• [61443761, 691443761; 8].

• [61803227, 618030227; 8].• [62387573, 623087573; 8].• [63499019, 634399019; 8].• [63998771, 639698771; 8].• [64658683, 634658683; 8].

• [65624953, 965624953; 8].• [65634313, 656343313; 8].• [65634313, 656343313; 8].• [65974387, 659743807; 8].• [67673087, 676736087; 8].

• [67753937, 677539307; 8].• [68334797, 683347937; 8].• [69220873, 692290873; 8].• [69228619, 692298619; 8].• [69636797, 696369797; 8].

• [70271167, 702711367; 8].• [70283047, 970283047; 8].• [70843057, 708403057; 8].• [71232599, 701232599; 8].• [71422609, 714226069; 8].

17


• [72893731, 728937331; 8].• [72893731, 728937331; 8].• [73274101, 732741091; 8].• [73966181, 739669181; 8].• [74459521, 744059521; 8].• [74483569, 704483569; 8].• [74901019, 749301019; 8].• [75277217, 705277217; 8].• [75373517, 753673517; 8].• [76544849, 765443849; 8].• [76659631, 766959631; 8].• [77018411, 770184131; 8].• [77089589, 770895989; 8].• [77769707, 737769707; 8].• [79648571, 709648571; 8].• [79955587, 793955587; 8].• [80020219, 800320219; 8].

• [80127911, 801267911; 8].• [81064943, 810364943; 8].• [82089803, 820890803; 8].• [82682351, 802682351; 8].• [84085031, 840850313; 8].• [84457229, 804457229; 8].• [89109773, 891097673; 8].• [89381297, 893812697; 8].• [90090197, 909090197; 8].• [90195389, 901956389; 8].• [91097719, 910977619; 8].• [91119209, 911192099; 8].• [91119209, 911192099; 8].• [91814617, 918143617; 8].• [92367841, 923678641; 8].• [92805337, 928053037; 8].• [92940527, 929405207; 8].

• [93120667, 931206607; 8].• [93772769, 937727069; 8].• [94128253, 994128253; 8].• [94128253, 994128253; 8].• [94176941, 934176941; 8].• [94227703, 934227703; 8].• [94244977, 964244977; 8].• [94536511, 945360511; 8].• [94742399, 947342399; 8].• [95002891, 956002891; 8].• [95469109, 954691039; 8].• [96966197, 969661907; 8].• [98359057, 980359057; 8].• [98662873, 980662873; 8].• [99561373, 995613793; 8].• [99905431, 939905431; 8].

4.2 Two Digits RepetitionsBelow are few examples, with simplified notations. The first example 1709 can also be seen at [3] butonly with 57. While we have done it in two ways, 57 and 75. See below:

• [1709, 175709; 8] I 1 7091 75 7091 75 75 7091 75 75 75 7091 75 75 75 75 7091 75 75 75 75 75 7091 75 75 75 75 75 75 7091 75 75 75 75 75 75 75 709.

• [1709, 175709; 8] I 17 0917 57 0917 57 57 0917 57 57 57 0917 57 57 57 57 0917 57 57 57 57 57 0917 57 57 57 57 57 57 0917 57 57 57 57 57 57 57 09.


• [14843, 7214843; 8].• [65449, 6335449; 8].

• [107791, 78107791; 8].• [176933, 17693993; 8].• [184517, 18694517; 8].

• [297907, 29790763; 8].• [368111, 36860111; 8].• [374287, 37460287; 8].• [627797, 62967797; 8].• [718183, 27718183; 8].

• [726697, 72336697; 8].• [787807, 66787807; 8].• [793621, 78493621; 8].• [848131, 84813451; 8].• [858149, 85908149; 8].• [882877, 83382877; 8].

18


• [1070011, 196070011; 8].• [1136053, 113609953; 8].• [1502407, 150246007; 8].• [1562947, 156782947; 8].• [1903747, 151903747; 8].

• [1903747, 151903747; 8].• [1931213, 631931213; 8].• [2182021, 332182021; 8].• [2205823, 220582393; 8].• [2205823, 220582393; 8].

• [2416133, 572416133; 8].• [2702059, 270209959; 8].• [2703013, 293703013; 8].• [2717249, 271397249; 8].• [2738947, 273887947; 8].

• [2860673, 422860673; 8].• [3406223, 340622273; 8].• [3756023, 375602243; 8].• [3862841, 386332841; 8].• [3919319, 843919319; 8].

• [3993029, 399307229; 8].• [4102123, 410254123; 8].• [4174561, 417427561; 8].• [4515113, 421515113; 8].• [4999591, 964999591; 8].

• [5102381, 510278381; 8].• [5167637, 516763427; 8].• [5272447, 527244727; 8].• [5272447, 527244727; 8].• [5703899, 570399899; 8].

• [5858113, 585158113; 8].• [5858113, 585158113; 8].• [6126203, 612620633; 8].• [6225701, 654225701; 8].• [6399797, 639489797; 8].

• [6440113, 654440113; 8].• [6531391, 653139241; 8].• [6642847, 664284217; 8].• [6754147, 756754147; 8].• [6833549, 683213549; 8].

• [7023173, 702366173; 8].• [7105699, 710569159; 8].• [7226357, 722635757; 8].• [7226357, 722635757; 8].• [7226357, 722635757; 8].

• [7241519, 718241519; 8].• [7282519, 730282519; 8].• [7350373, 721350373; 8].• [7480307, 748030487; 8].• [7560599, 756990599; 8].

• [7888877, 788668877; 8].• [8268349, 833268349; 8].• [8357537, 835751237; 8].• [8371009, 837100339; 8].• [8476189, 884476189; 8].

• [9195757, 919575667; 8].• [9221857, 975221857; 8].• [9499579, 942199579; 8].• [9615439, 961543909; 8].• [9738793, 973857793; 8].

• [10190683, 1019068513; 8].• [10329437, 1032943697; 8].• [10563719, 1056351719; 8].• [10573181, 1057513181; 8].• [10892633, 1089263993; 8].

• [10957183, 1960957183; 8].• [11542207, 1154299207; 8].• [12367067, 1236706967; 8].• [12367067, 1236706967; 8].• [12563851, 1253363851; 8].

• [12630587, 1267530587; 8].• [13106633, 1310618633; 8].• [13178351, 1327178351; 8].• [13333273, 1333323373; 8].• [13425889, 1183425889; 8].

• [13445189, 1344518159; 8].• [13464643, 1393464643; 8].• [13464643, 1393464643; 8].• [13655263, 1365526339; 8].• [14010307, 3314010307; 8].

• [14083969, 1408393669; 8].• [14138323, 8114138323; 8].• [14488741, 1448874169; 8].• [14720059, 1472009659; 8].• [14750221, 1433750221; 8].

• [15225031, 1522501531; 8].• [15245033, 1524503993; 8].• [16083307, 1642083307; 8].• [16270627, 1156270627; 8].• [16330879, 1633087957; 8].

• [16545553, 1666545553; 8].• [16545553, 1666545553; 8].• [17023277, 1702183277; 8].• [17617973, 1761847973; 8].• [18061597, 1866061597; 8].

• [18259559, 1825959959; 8].• [18260377, 1308260377; 8].• [18557629, 1853657629; 8].• [18632113, 1908632113; 8].• [18653213, 1908653213; 8].

• [18856141, 1885611841; 8].• [18923153, 1892315351; 8].• [18929231, 1892669231; 8].• [18943381, 1894337281; 8].• [19086101, 1908610181; 8].

19


• [19086101, 1908610181; 8].• [20485121, 2048515421; 8].• [20602739, 2060276639; 8].• [20708183, 2075108183; 8].• [20894911, 2089491721; 8].

• [21125107, 2112515107; 8].• [21125107, 2112515107; 8].• [21125107, 2112515107; 8].• [21277609, 2127760279; 8].• [21336137, 2135736137; 8].

• [21448793, 2144871293; 8].• [21453191, 2139453191; 8].• [21551357, 1821551357; 8].• [21676667, 2871676667; 8].• [21714863, 2171482763; 8].

• [21772379, 4521772379; 8].• [21846281, 2172846281; 8].• [22156307, 2248156307; 8].• [22332677, 2233267427; 8].• [22378141, 2230378141; 8].

• [22542623, 2254233623; 8].• [22737877, 2273787847; 8].• [22846289, 2290846289; 8].• [22919513, 2291956613; 8].• [23165063, 2316506453; 8].

• [23200523, 2320052993; 8].• [23231083, 2326031083; 8].• [23252081, 2325263081; 8].• [23437229, 2343722999; 8].• [23437229, 2343722999; 8].

• [23694257, 2369664257; 8].• [24115879, 2411589979; 8].• [24168643, 2844168643; 8].• [24178409, 2417846309; 8].• [24257843, 2425784363; 8].

• [24257843, 2425784363; 8].• [24352717, 2435271187; 8].• [24522877, 2452284277; 8].• [24615377, 2334615377; 8].• [25107757, 2510775733; 8].

• [25240091, 2995240091; 8].• [25567693, 2533567693; 8].• [25636873, 2569636873; 8].• [25636873, 2569636873; 8].• [25741847, 2574199847; 8].

• [25891003, 2935891003; 8].• [25923041, 2592333041; 8].• [25923041, 2592333041; 8].• [26213389, 2621351389; 8].• [26510509, 2651840509; 8].

• [27120019, 1227120019; 8].• [27121649, 2712301649; 8].• [27735863, 2773551863; 8].• [27967531, 2337967531; 8].• [27980669, 2792180669; 8].

• [28023007, 7228023007; 8].• [28035143, 2803512443; 8].• [28058861, 2338058861; 8].• [28112569, 2811256933; 8].• [28288829, 2828869829; 8].

• [28631059, 5728631059; 8].• [28716241, 2758716241; 8].• [28821301, 2836821301; 8].• [29078249, 2907788249; 8].• [29208793, 2920812793; 8].

• [29306611, 2935106611; 8].• [29370983, 2339370983; 8].• [29515271, 2951527133; 8].• [29569447, 2956579447; 8].• [29718583, 2993718583; 8].

• [29732771, 2973277421; 8].• [30082649, 3008264933; 8].• [30155003, 3015500483; 8].• [30199009, 3019909309; 8].• [30343259, 1230343259; 8].

• [30645313, 3039645313; 8].• [31168777, 3116873077; 8].• [31316713, 3133316713; 8].• [31316713, 3133316713; 8].• [31432831, 3541432831; 8].

• [31693591, 3169353391; 8].• [31771361, 3177175361; 8].• [31812983, 9331812983; 8].• [31943411, 3194341391; 8].• [32673821, 3512673821; 8].

• [32903911, 3152903911; 8].• [33187079, 3318127079; 8].• [33379547, 3338179547; 8].• [33711317, 3371631317; 8].• [34523351, 3442523351; 8].

• [34542187, 3499542187; 8].• [34689749, 3468933749; 8].• [34699507, 3469954207; 8].• [34791307, 8134791307; 8].• [35431717, 9935431717; 8].

• [35797939, 3579793849; 8].• [35897207, 6335897207; 8].• [35910781, 3592410781; 8].• [36035473, 3603395473; 8].• [36401059, 2436401059; 8].

• [36772633, 3677263903; 8].• [37041383, 3704169383; 8].• [37181173, 3718193173; 8].• [37230269, 3721530269; 8].• [37553297, 3772553297; 8].

20


• [37668401, 4837668401; 8].• [37771529, 3777901529; 8].• [38215153, 3821512153; 8].• [38215153, 3821512153; 8].• [38677411, 3867457411; 8].

• [38726869, 3872686369; 8].• [38726869, 3872686369; 8].• [39346949, 3934694759; 8].• [39401083, 3940108813; 8].• [39435287, 3943584287; 8].

• [39497057, 3972497057; 8].• [39546667, 3954669067; 8].• [40256387, 4025638427; 8].• [41274089, 4451274089; 8].• [41459153, 4145915351; 8].

• [41849677, 4139849677; 8].• [42233413, 4299233413; 8].• [42267331, 4226727331; 8].• [42267331, 4226727331; 8].• [43075859, 4330075859; 8].

• [43523219, 4353023219; 8].• [43597969, 4356997969; 8].• [43874329, 4387814329; 8].• [43876919, 8743876919; 8].• [43977169, 4399077169; 8].

• [43996453, 4369996453; 8].• [44063549, 4406354699; 8].• [44067293, 4466067293; 8].• [44104531, 4410453121; 8].• [44104531, 4410453121; 8].

• [44215937, 4421598437; 8].• [44752447, 4475842447; 8].• [44845019, 6044845019; 8].• [45027007, 4502701807; 8].• [45287147, 1545287147; 8].

• [45297559, 4529715559; 8].• [45476729, 4547672219; 8].• [45966407, 4596640751; 8].• [46156477, 4633156477; 8].• [46336819, 4516336819; 8].

• [46583371, 4658603371; 8].• [46670033, 4667120033; 8].• [46804721, 4996804721; 8].• [47686421, 4768642133; 8].• [47796233, 4779684233; 8].

• [47800261, 4780066261; 8].• [47822587, 4782255487; 8].• [48398761, 4839876193; 8].• [48403967, 4840393967; 8].• [48403967, 4840393967; 8].

• [48403967, 4840393967; 8].• [48413051, 4841366051; 8].• [48448039, 4844803219; 8].• [48649399, 4864669399; 8].• [48710887, 4879010887; 8].

• [48770669, 4857770669; 8].• [49000993, 2149000993; 8].• [49126579, 4912659979; 8].• [49297697, 4954297697; 8].• [49607039, 9949607039; 8].

• [49640753, 4964066753; 8].• [49682533, 4984682533; 8].• [50656819, 5065936819; 8].• [50672987, 5062172987; 8].• [50728339, 5072833999; 8].

• [50728339, 5072833999; 8].• [50770571, 5390770571; 8].• [50904839, 5066904839; 8].• [51240961, 5124096751; 8].• [51338321, 5136038321; 8].

• [51773237, 8451773237; 8].• [51933961, 5163933961; 8].• [51947507, 5194750733; 8].• [51969109, 5196429109; 8].• [52204753, 5220493753; 8].

• [52722689, 3052722689; 8].• [52793099, 1852793099; 8].• [53058667, 5305867867; 8].• [53099729, 5345099729; 8].• [53168041, 5316788041; 8].

• [53429393, 5342936693; 8].• [53454563, 5345456513; 8].• [53690759, 5369075549; 8].• [53878333, 5345878333; 8].• [54193829, 5419389629; 8].

• [54309757, 5430975577; 8].• [54436793, 5443679357; 8].• [54624439, 5468724439; 8].• [54878123, 5487368123; 8].• [54922177, 5499922177; 8].

• [54922177, 5499922177; 8].• [55151267, 5515123967; 8].• [55197047, 5519709347; 8].• [55311203, 5531133203; 8].• [55344977, 5599344977; 8].

• [55440313, 5544031393; 8].• [55440313, 5544031393; 8].• [55509427, 5550966427; 8].• [55857749, 5335857749; 8].• [56150309, 5615030969; 8].

• [56150309, 5615030969; 8].• [56191301, 5186191301; 8].• [56281859, 5628185279; 8].• [56904539, 5690844539; 8].• [57100471, 5710099471; 8].

21


• [57262853, 5726285333; 8].• [57262853, 5726285333; 8].• [57393451, 5732493451; 8].• [57455999, 5745425999; 8].• [57575461, 5733575461; 8].

• [57732439, 5307732439; 8].• [57804191, 5785704191; 8].• [57815213, 9657815213; 8].• [58335983, 5833542983; 8].• [58562741, 5836562741; 8].

• [58587839, 5218587839; 8].• [58801073, 5218801073; 8].• [58858439, 5885843549; 8].• [59021251, 5905421251; 8].• [59058617, 5924058617; 8].

• [59210029, 8459210029; 8].• [59467307, 5969467307; 8].• [59467307, 5969467307; 8].• [59559217, 5956359217; 8].• [59964977, 5996497937; 8].

• [59968837, 5995768837; 8].• [60497179, 6049577179; 8].• [60939617, 6093966617; 8].• [60939617, 6093966617; 8].• [61514977, 6151466977; 8].

• [62276057, 6263276057; 8].• [62946353, 6294636353; 8].• [62946353, 6294636353; 8].• [62946353, 6294636353; 8].• [62981837, 6298183967; 8].

• [63285191, 6328519157; 8].• [63379751, 6337973951; 8].• [63508199, 6372508199; 8].• [64091119, 6409601119; 8].• [64298161, 6429842161; 8].

• [64970359, 6491570359; 8].• [65409271, 6185409271; 8].• [65446001, 6544603301; 8].• [65504027, 3965504027; 8].• [65779933, 6785779933; 8].

• [65836163, 6586636163; 8].• [65886949, 6588690949; 8].• [66471631, 6647167231; 8].• [66506267, 6306506267; 8].• [66818317, 6681833317; 8].

• [66818317, 6681833317; 8].• [66911417, 6691133417; 8].• [67020559, 6702055849; 8].• [67100497, 6714200497; 8].• [67346003, 9967346003; 8].

• [67392077, 9367392077; 8].• [67396657, 6739663657; 8].• [67396657, 6739663657; 8].• [67617863, 6766617863; 8].• [67617863, 6766617863; 8].

• [67707803, 6770783603; 8].• [67789493, 6997789493; 8].• [67932791, 6793274591; 8].• [68785037, 9968785037; 8].• [68912863, 6891286387; 8].

• [69288091, 6915288091; 8].• [69636247, 6951636247; 8].• [69746309, 6974630339; 8].• [69796187, 6971296187; 8].• [70046609, 7004660339; 8].

• [70325401, 7039925401; 8].• [70351111, 7035142111; 8].• [70358279, 7035836279; 8].• [70569259, 7056921259; 8].• [70569259, 7056921259; 8].

• [70621223, 7330621223; 8].• [70663157, 7066375157; 8].• [70886203, 7088633203; 8].• [71091079, 7109184079; 8].• [71108183, 7110998183; 8].

• [71704421, 7170964421; 8].• [71745977, 7174560977; 8].• [71919359, 7191939059; 8].• [72030397, 7287030397; 8].• [72191381, 7217891381; 8].

• [72275851, 7227585133; 8].• [72392399, 7239299399; 8].• [72459731, 1872459731; 8].• [72720041, 7782720041; 8].• [72890963, 7332890963; 8].

• [73593293, 7423593293; 8].• [73617559, 7361755339; 8].• [73631911, 7363191133; 8].• [74440609, 3974440609; 8].• [74604749, 7460474933; 8].

• [74612903, 7461297803; 8].• [74663899, 7466389369; 8].• [74789531, 5174789531; 8].• [74899777, 7489981777; 8].• [74960617, 7436960617; 8].

• [75891089, 7589841089; 8].• [76242581, 7626642581; 8].• [76331551, 1876331551; 8].• [76910767, 7693910767; 8].• [76910767, 7693910767; 8].

• [76947847, 7675947847; 8].• [77187373, 7718737153; 8].• [77279537, 7727953367; 8].• [77884907, 7789984907; 8].• [78352091, 6678352091; 8].

22


• [78670387, 7867570387; 8].• [78670387, 7867570387; 8].• [78673481, 7869073481; 8].• [78821341, 7882169341; 8].• [78914981, 7758914981; 8].• [78962287, 7896229687; 8].• [79592951, 7959295121; 8].• [79592951, 7959295121; 8].• [79685083, 7975685083; 8].• [81195463, 8119155463; 8].

• [81232331, 8123236631; 8].• [81331717, 8133133717; 8].• [82244947, 8224154947; 8].• [82630313, 8218630313; 8].• [82778149, 8277668149; 8].• [82791301, 8452791301; 8].• [82902179, 8298402179; 8].• [83061359, 8306135951; 8].• [83578591, 3083578591; 8].• [84040973, 8404990973; 8].• [84623041, 8462309941; 8].• [84979369, 1284979369; 8].• [85095683, 8509566683; 8].• [85095683, 8509566683; 8].• [85173331, 8517663331; 8].• [85318421, 8995318421; 8].• [85350367, 8535036817; 8].

• [85630109, 8533630109; 8].• [85677703, 4585677703; 8].• [85778687, 8573378687; 8].• [85855349, 8585815349; 8].• [85930129, 8815930129; 8].• [85977317, 8597778317; 8].• [86833661, 8683123661; 8].• [86985979, 8306985979; 8].

• [87360083, 8738460083; 8].• [87645457, 8764530457; 8].• [88175557, 3088175557; 8].• [88227239, 8822457239; 8].• [88320019, 8839020019; 8].• [88329089, 8832908219; 8].• [88690141, 8869019941; 8].• [89587991, 8958697991; 8].• [89766221, 8759766221; 8].• [90129619, 9180129619; 8].

• [90273331, 9027369331; 8].• [90471779, 9330471779; 8].• [90582803, 9059982803; 8].• [90612607, 9061260517; 8].• [90728731, 9072338731; 8].• [90978253, 9097896253; 8].• [91177753, 9117775243; 8].

• [91193243, 9119324453; 8].• [91449719, 6691449719; 8].• [91510411, 9169510411; 8].• [91961179, 5791961179; 8].• [92556287, 9284556287; 8].• [92892073, 9289932073; 8].• [93053503, 9305358103; 8].• [93483319, 9348331957; 8].• [93738371, 9373398371; 8].• [93818987, 9381897587; 8].• [93897161, 9389713661; 8].• [94232557, 9423812557; 8].• [94385143, 9438514303; 8].

• [94483861, 3394483861; 8].• [94602367, 9460452367; 8].• [95185457, 9511885457; 8].• [95288813, 9528908813; 8].• [95551217, 9555128417; 8].• [96483323, 9648243323; 8].• [97610783, 9761076683; 8].• [97681211, 8197681211; 8].• [97921337, 9792178337; 8].• [98108623, 9810862183; 8].• [98136229, 9899136229; 8].• [98395043, 9821395043; 8].• [98456053, 9845687053; 8].• [99504061, 9950814061; 8].

4.3 Three Digits RepetitionsBelow are few examples, with simplified notation.

• [281, 213281; 8] I 281213 281213 213 281213 213 213 281213 213 213 213 281213 213 213 213 213 281213 213 213 213 213 213 281213 213 213 213 213 213 213 281.

• [281, 213281; 8] I 2 812 132 812 132 132 812 132 132 132 812 132 132 132 132 812 132 132 132 132 132 812 132 132 132 132 132 132 812 132 132 132 132 132 132 132 81.

23



• [3109, 1533109; 8].• [3527, 3273527; 8].• [3527, 3273527; 8].• [7639, 2827639; 8].

• [23831, 23866331; 8].• [36871, 33936871; 8].• [36871, 33936871; 8].• [41057, 41264057; 8].• [45481, 45482221; 8].• [51803, 75051803; 8].

• [54101, 54170801; 8].• [56369, 56324069; 8].• [68897, 68606897; 8].• [79769, 79906769; 8].• [80329, 80329933; 8].• [83357, 83228357; 8].

• [111269, 119601269; 8].• [117883, 117883327; 8].• [152899, 153962899; 8].• [160253, 161290253; 8].• [206081, 282806081; 8].

• [240173, 240128873; 8].• [245711, 245769311; 8].• [248309, 248309069; 8].• [249521, 249486521; 8].• [264301, 283464301; 8].

• [280219, 582280219; 8].• [300301, 300301153; 8].• [300661, 304170661; 8].• [301913, 308221913; 8].• [325201, 325295101; 8].

• [326693, 326672693; 8].• [326693, 326672693; 8].• [332951, 332141951; 8].• [361549, 105361549; 8].• [383549, 383541089; 8].

• [395509, 367895509; 8].• [404843, 404375843; 8].• [407311, 407375911; 8].• [423853, 423189853; 8].• [444811, 444818611; 8].

• [444811, 444818611; 8].• [457277, 457275827; 8].• [465931, 312465931; 8].• [476783, 484076783; 8].• [486481, 486488161; 8].

• [488153, 488939153; 8].• [488321, 488389721; 8].• [497171, 495257171; 8].• [499189, 897499189; 8].• [500009, 500237009; 8].

• [505691, 505687691; 8].• [505691, 505687691; 8].• [506213, 171506213; 8].• [510319, 306510319; 8].• [510383, 564510383; 8].

• [510383, 564510383; 8].• [510817, 510876217; 8].• [519097, 518169097; 8].• [520021, 520022611; 8].• [528791, 581628791; 8].

• [543887, 543884777; 8].• [544717, 541174717; 8].• [550859, 550726859; 8].• [569111, 569194511; 8].• [579713, 579792713; 8].

• [579713, 579792713; 8].• [610921, 699310921; 8].• [643651, 649813651; 8].• [644327, 649124327; 8].• [652207, 652207123; 8].

• [655547, 655945547; 8].• [655547, 655945547; 8].• [697993, 697995313; 8].• [703123, 703129033; 8].• [714227, 714708227; 8].

• [731327, 736841327; 8].• [795763, 855795763; 8].• [824477, 865724477; 8].• [833887, 833829487; 8].

• [839653, 838799653; 8].• [845137, 909845137; 8].• [846931, 846966331; 8].• [860071, 860345071; 8].

• [862559, 862533059; 8].• [867623, 867714623; 8].• [873251, 876933251; 8].• [907099, 907573099; 8].• [943589, 947023589; 8].• [974273, 974273471; 8].• [989783, 279989783; 8].

• [1012373, 1012373933; 8].• [1012373, 1012373933; 8].• [1021093, 1021066093; 8].• [1035263, 1882035263; 8].• [1041163, 1333041163; 8].

• [1057249, 1057324249; 8].• [1066619, 1066501619; 8].• [1087271, 1087230071; 8].• [1105547, 1990105547; 8].• [1117673, 1117677263; 8].

24


• [1140341, 1140341483; 8].• [1161047, 3961161047; 8].• [1163333, 1261163333; 8].• [1163333, 1261163333; 8].• [1174783, 1306174783; 8].

• [1208663, 1208690363; 8].• [1210393, 1210399513; 8].• [1232527, 1232585227; 8].• [1234819, 1234838119; 8].• [1256809, 1256612809; 8].

• [1280597, 1283090597; 8].• [1322437, 1322460637; 8].• [1350809, 1354680809; 8].• [1361959, 1361977159; 8].• [1381229, 1381855229; 8].

• [1400479, 1400693479; 8].• [1407143, 1407138743; 8].• [1426801, 1426848001; 8].• [1452991, 1726452991; 8].• [1458727, 1490058727; 8].

• [1461809, 1705461809; 8].• [1474313, 1474549313; 8].• [1506619, 1506456619; 8].• [1506619, 1506456619; 8].• [1515847, 1519635847; 8].

• [1520723, 1520719523; 8].• [1552879, 1558342879; 8].• [1574467, 1574467519; 8].• [1595557, 1517195557; 8].• [1630463, 1606630463; 8].

• [1630927, 1677730927; 8].• [1648253, 1646218253; 8].• [1650353, 1651020353; 8].• [1685917, 1369685917; 8].• [1732397, 1732398917; 8].

• [1751773, 1751771713; 8].• [1768667, 1768667657; 8].• [1768667, 1768667657; 8].• [1779881, 1758579881; 8].• [1783069, 1783089169; 8].

• [1787587, 1787583757; 8].• [1808773, 1808708773; 8].• [1808773, 1808708773; 8].• [1810481, 1810484021; 8].• [1813319, 1816543319; 8].

• [1832057, 1832054597; 8].• [1834193, 1873534193; 8].• [1842847, 4861842847; 8].• [1859519, 1273859519; 8].• [1866941, 1833666941; 8].

• [1880647, 1818380647; 8].• [1884679, 4981884679; 8].• [1904509, 1904552809; 8].• [1935149, 1976535149; 8].• [1944323, 1944321653; 8].

• [1947527, 1948437527; 8].• [1961833, 1964591833; 8].• [1966043, 6091966043; 8].• [1979303, 1979371403; 8].• [1982779, 1982771929; 8].

• [2006441, 2102006441; 8].• [2006441, 2102006441; 8].• [2010167, 2022210167; 8].• [2073469, 2073463969; 8].• [2073469, 2073463969; 8].

• [2097829, 2072097829; 8].• [2097829, 2072097829; 8].• [2117389, 2117870389; 8].• [2145643, 2135745643; 8].• [2172791, 2178882791; 8].

• [2173183, 1052173183; 8].• [2177899, 2127977899; 8].• [2209997, 2209997147; 8].• [2209997, 2209997147; 8].• [2212081, 2212081219; 8].

• [2213143, 2262413143; 8].• [2241311, 2241348911; 8].• [2298379, 9692298379; 8].• [2305747, 2307775747; 8].• [2326613, 2326837613; 8].

• [2331419, 2331948419; 8].• [2343521, 2343360521; 8].• [2356141, 2358076141; 8].• [2380877, 2385370877; 8].• [2420269, 2420606269; 8].

• [2456621, 2483756621; 8].• [2460113, 2466180113; 8].• [2475061, 2475067261; 8].• [2475061, 2475067261; 8].• [2497661, 2490997661; 8].

• [2506489, 2471506489; 8].• [2538791, 1292538791; 8].• [2560661, 2560213661; 8].• [2597489, 2597481329; 8].• [2631691, 2637411691; 8].

• [2635037, 2635040237; 8].• [2652373, 2652373621; 8].• [2670191, 2673720191; 8].• [2694907, 2315694907; 8].• [2709599, 2705199599; 8].

• [2717563, 1382717563; 8].• [2722949, 2722944899; 8].• [2733799, 2733719299; 8].• [2752823, 2752570823; 8].• [2768123, 2743268123; 8].

25


• [2775989, 2775940289; 8].• [2781073, 2781039973; 8].• [2781853, 2781852223; 8].• [2844239, 2132844239; 8].• [2844239, 2132844239; 8].

• [2849779, 2849774329; 8].• [2860751, 2860942751; 8].• [2865509, 2993865509; 8].• [2883851, 2883859061; 8].• [2884073, 2738884073; 8].

• [2900621, 2907860621; 8].• [2976689, 2964576689; 8].• [2994113, 2994116933; 8].• [3004319, 3003394319; 8].• [3012719, 3012724919; 8].

• [3027749, 3027288749; 8].• [3032399, 3086732399; 8].• [3043217, 8673043217; 8].• [3043739, 3038143739; 8].• [3154297, 3154366297; 8].

• [3165287, 3168945287; 8].• [3173069, 3177143069; 8].• [3199591, 3199417591; 8].• [3232517, 3232489517; 8].• [3270313, 3270356413; 8].

• [3283879, 3150283879; 8].• [3294631, 8853294631; 8].• [3295657, 3295657951; 8].• [3330527, 3330927527; 8].• [3332281, 3332600281; 8].

• [3349163, 3349163567; 8].• [3351301, 5403351301; 8].• [3360689, 3360686129; 8].• [3379249, 8823379249; 8].• [3380833, 3380681833; 8].

• [3404081, 3404099381; 8].• [3415813, 3415264813; 8].• [3416939, 3297416939; 8].• [3440147, 3440447147; 8].• [3448507, 1893448507; 8].

• [3470029, 3470029357; 8].• [3496613, 3496618913; 8].• [3496613, 3496618913; 8].• [3498709, 3498722209; 8].• [3514729, 3514573729; 8].

• [3547519, 3551347519; 8].• [3548177, 3548143877; 8].• [3571793, 3571785293; 8].• [3587687, 3587684297; 8].• [3607921, 3607921621; 8].

• [3607921, 3607921621; 8].• [3607921, 3607921621; 8].• [3638203, 3635288203; 8].• [3642707, 3642701057; 8].• [3657449, 8193657449; 8].

• [3671089, 3105671089; 8].• [3712091, 3712022291; 8].• [3713219, 3713278619; 8].• [3714731, 3714731843; 8].• [3744017, 3744017489; 8].

• [3750203, 3750276203; 8].• [3750203, 3750276203; 8].• [3809081, 3372809081; 8].• [3814093, 3814094833; 8].• [3847849, 3762847849; 8].

• [3867001, 3887667001; 8].• [3889183, 3889155883; 8].• [3907121, 3907128161; 8].• [3943661, 3943667111; 8].• [3955771, 3955144771; 8].

• [3982129, 3971182129; 8].• [4001791, 4001303791; 8].• [4031221, 1024031221; 8].• [4031761, 4047731761; 8].• [4040677, 4032440677; 8].

• [4104461, 4246104461; 8].• [4132273, 4132293973; 8].• [4185821, 4185827231; 8].• [4191151, 4191151843; 8].• [4191443, 4191442133; 8].

• [4197883, 4193277883; 8].• [4208759, 4208759453; 8].• [4221433, 4084221433; 8].• [4246751, 4246765151; 8].• [4247069, 4256747069; 8].

• [4258139, 4258264139; 8].• [4275781, 4354275781; 8].• [4275781, 4354275781; 8].• [4295443, 4292735443; 8].• [4318469, 4318441769; 8].

• [4341277, 4332741277; 8].• [4396463, 4396434863; 8].• [4399333, 4690399333; 8].• [4439453, 4912439453; 8].• [4442839, 4442833999; 8].

• [4477453, 3784477453; 8].• [4487339, 4487327639; 8].• [4512397, 4360512397; 8].• [4531847, 4531843247; 8].• [4531847, 4531843247; 8].

• [4537579, 4548637579; 8].• [4539481, 4564839481; 8].• [4541501, 4542251501; 8].• [4554083, 4554081473; 8].• [4587211, 4584387211; 8].

26


• [4587211, 4584387211; 8].• [4608311, 4608939311; 8].• [4619213, 4619232413; 8].• [4630949, 4656430949; 8].• [4638187, 4638657187; 8].

• [4651837, 2794651837; 8].• [4652117, 4652330117; 8].• [4688143, 4688128843; 8].• [4695623, 4695591623; 8].• [4753613, 4753613633; 8].

• [4753613, 4753613633; 8].• [4754627, 4754645627; 8].• [4754627, 4754645627; 8].• [4768873, 1654768873; 8].• [4829663, 4817129663; 8].

• [4845461, 4845075461; 8].• [4886461, 4777886461; 8].• [4931183, 4931558183; 8].• [4933289, 4936483289; 8].• [4935103, 4935112603; 8].

• [4970699, 4970291699; 8].• [4991953, 4795991953; 8].• [4995017, 4969395017; 8].• [5050681, 5050643881; 8].• [5054173, 5053484173; 8].

• [5060833, 5060837533; 8].• [5060833, 5060837533; 8].• [5067263, 5067254963; 8].• [5118667, 5118624967; 8].• [5128909, 5110828909; 8].

• [5130553, 9005130553; 8].• [5173859, 5173859219; 8].• [5173859, 5173859219; 8].• [5178521, 5162178521; 8].• [5178521, 5162178521; 8].

• [5180099, 5180270099; 8].• [5206829, 5206829351; 8].• [5211389, 5218851389; 8].• [5255843, 5255855243; 8].• [5278927, 5272288927; 8].

• [5283809, 5299983809; 8].• [5307143, 5360307143; 8].• [5307143, 5360307143; 8].• [5319737, 5319738707; 8].• [5371199, 5371196399; 8].

• [5371199, 5371196399; 8].• [5375011, 5675375011; 8].• [5375011, 5675375011; 8].• [5392489, 5395822489; 8].• [5432731, 5432734621; 8].

• [5438137, 5438723137; 8].• [5455897, 5486155897; 8].• [5467097, 3515467097; 8].• [5495383, 5495386513; 8].• [5534813, 5534898713; 8].

• [5535401, 5535402161; 8].• [5539883, 5539887173; 8].• [5579573, 5657579573; 8].• [5592077, 5592077609; 8].• [5602529, 5602593629; 8].

• [5644127, 5613544127; 8].• [5664013, 5664600013; 8].• [5715209, 6095715209; 8].• [5743651, 5743659811; 8].• [5789869, 5722889869; 8].

• [5794507, 5794567507; 8].• [5794507, 5794567507; 8].• [5798189, 5687798189; 8].• [5812483, 5898712483; 8].• [5871301, 5872971301; 8].

• [5871301, 5872971301; 8].• [5889517, 5842389517; 8].• [6001013, 6057301013; 8].• [6013309, 6035113309; 8].• [6050047, 6050065447; 8].

• [6071581, 6071567581; 8].• [6071581, 6071567581; 8].• [6077371, 6017177371; 8].• [6101621, 6178301621; 8].• [6105179, 6939105179; 8].

• [6192871, 6894192871; 8].• [6204097, 6204070597; 8].• [6218117, 6218113637; 8].• [6230759, 6230757389; 8].• [6251797, 6251791747; 8].

• [6265783, 6272065783; 8].• [6267907, 6267908827; 8].• [6303001, 6303001867; 8].• [6310727, 7986310727; 8].• [6325283, 6313825283; 8].

• [6338107, 4146338107; 8].• [6354391, 6273354391; 8].• [6358351, 6358858351; 8].• [6358351, 6358858351; 8].• [6358351, 6358858351; 8].

• [6371929, 6252371929; 8].• [6374213, 6377354213; 8].• [6375067, 6885375067; 8].• [6440771, 6440772131; 8].• [6481177, 6481177849; 8].

• [6484319, 6471184319; 8].• [6488683, 6488609683; 8].• [6518003, 6600518003; 8].• [6525457, 6561225457; 8].• [6534197, 6534192767; 8].

27


• [6556609, 5136556609; 8].• [6562001, 6159562001; 8].• [6606101, 6606669101; 8].• [6626561, 6628916561; 8].• [6629807, 6629867207; 8].

• [6636199, 2646636199; 8].• [6646867, 8316646867; 8].• [6658159, 6801658159; 8].• [6682573, 6682573207; 8].• [6688673, 7416688673; 8].

• [6768793, 6784068793; 8].• [6816617, 6821016617; 8].• [6842417, 6845072417; 8].• [6879673, 4836879673; 8].• [6891637, 6891631837; 8].

• [6891637, 6891631837; 8].• [6910499, 5076910499; 8].• [6915499, 6918015499; 8].• [6915499, 6918015499; 8].• [6923197, 6923783197; 8].

• [6923197, 6923783197; 8].• [6951047, 6453951047; 8].• [6951583, 6306951583; 8].• [6951583, 6306951583; 8].• [6980189, 6980777189; 8].

• [6990799, 6990633799; 8].• [7040041, 7024640041; 8].• [7053127, 7016853127; 8].• [7059883, 7059768883; 8].• [7106503, 7708106503; 8].

• [7126279, 7126789279; 8].• [7134137, 7134945137; 8].• [7141427, 7141427369; 8].• [7249399, 1867249399; 8].• [7270663, 7270342663; 8].

• [7272283, 1387272283; 8].• [7312087, 7312606087; 8].• [7335997, 7335735997; 8].• [7335997, 7335735997; 8].• [7335997, 7335735997; 8].

• [7346827, 3277346827; 8].• [7351219, 7351678219; 8].• [7369639, 7384369639; 8].• [7369639, 7384369639; 8].• [7404421, 7912404421; 8].

• [7420121, 7135420121; 8].• [7444961, 7444645961; 8].• [7458533, 7458539573; 8].• [7469941, 7467299941; 8].• [7470499, 7472940499; 8].

• [7481387, 7421081387; 8].• [7485299, 7485299621; 8].• [7486459, 8887486459; 8].• [7525633, 7525635373; 8].• [7541129, 5917541129; 8].

• [7547263, 7132547263; 8].• [7579111, 7577899111; 8].• [7601837, 7601863037; 8].• [7623593, 7621383593; 8].• [7624657, 7162624657; 8].

• [7633559, 7633558799; 8].• [7644587, 3397644587; 8].• [7645237, 6217645237; 8].• [7645817, 9727645817; 8].• [7655789, 4207655789; 8].

• [7660811, 7660171811; 8].• [7670363, 6217670363; 8].• [7674311, 7674329111; 8].• [7712063, 8347712063; 8].• [7729019, 8617729019; 8].

• [7739759, 7739759351; 8].• [7788367, 7788380767; 8].• [7803721, 7807293721; 8].• [7818067, 7818067141; 8].• [7819753, 9847819753; 8].

• [7823833, 7855523833; 8].• [7883077, 7889753077; 8].• [7891259, 7390891259; 8].• [7905529, 7902525529; 8].• [8103149, 8149503149; 8].• [8134613, 8138494613; 8].

• [8147933, 8147949833; 8].• [8151763, 3068151763; 8].• [8219213, 8238419213; 8].• [8245417, 8238145417; 8].• [8258567, 8274758567; 8].• [8266931, 8265976931; 8].

• [8270617, 2468270617; 8].• [8277211, 8792277211; 8].• [8303623, 8303662723; 8].• [8328233, 1568328233; 8].• [8333861, 8333252861; 8].

• [8341631, 8380741631; 8].• [8354351, 8117354351; 8].• [8361137, 8361137699; 8].• [8361967, 8615361967; 8].• [8394223, 2438394223; 8].

• [8415457, 3518415457; 8].• [8418469, 8418276469; 8].• [8522429, 8524982429; 8].• [8528969, 8528918969; 8].• [8528969, 8528918969; 8].

28


• [8528969, 8528918969; 8].• [8540027, 8285540027; 8].• [8575969, 8954575969; 8].• [8611349, 8613781349; 8].• [8617237, 8675317237; 8].

• [8639177, 8639498177; 8].• [8642063, 1808642063; 8].• [8651267, 8663351267; 8].• [8659513, 8711659513; 8].• [8671907, 8678671907; 8].

• [8671907, 8678671907; 8].• [8671907, 8678671907; 8].• [8671907, 8678671907; 8].• [8672927, 8677412927; 8].• [8674891, 1868674891; 8].

• [8687911, 6338687911; 8].• [8703283, 1208703283; 8].• [8712701, 8726112701; 8].• [8719943, 5528719943; 8].• [8727287, 8727264587; 8].

• [8750867, 8750878067; 8].• [8755613, 8733955613; 8].• [8800303, 8800957303; 8].• [8803253, 8805703253; 8].• [8824733, 1388824733; 8].

• [8859863, 3248859863; 8].• [8872979, 8872971479; 8].• [8872979, 8872971479; 8].• [8915363, 8915365523; 8].• [8961679, 8961945679; 8].

• [9011213, 9011218913; 8].• [9011213, 9011218913; 8].• [9021241, 9082221241; 8].• [9033769, 4989033769; 8].• [9055601, 9507055601; 8].

• [9068119, 4929068119; 8].• [9076673, 9255076673; 8].• [9101899, 9101866399; 8].• [9107603, 9107604083; 8].• [9119113, 9198719113; 8].

• [9138179, 4059138179; 8].• [9149951, 9149910851; 8].• [9155723, 9155723033; 8].• [9182561, 9182567861; 8].• [9182561, 9182567861; 8].

• [9191233, 9191240533; 8].• [9196333, 9196333519; 8].• [9241777, 9242101777; 8].• [9252433, 9259392433; 8].• [9271891, 9271891201; 8].

• [9271891, 9271891201; 8].• [9301367, 9301462367; 8].• [9302257, 9302114257; 8].• [9318137, 9325218137; 8].• [9342583, 9342595183; 8].

• [9343909, 9957343909; 8].• [9388949, 9388982549; 8].• [9419749, 9495419749; 8].• [9419749, 9495419749; 8].• [9424501, 9422824501; 8].

• [9424501, 9422824501; 8].• [9491483, 9491488043; 8].• [9503939, 9501413939; 8].• [9536357, 9562736357; 8].• [9547337, 9547335947; 8].

• [9561691, 5259561691; 8].• [9565891, 9565561891; 8].• [9571489, 9571576489; 8].• [9620627, 9627350627; 8].• [9631709, 9611731709; 8].

• [9651833, 7899651833; 8].• [9716501, 4839716501; 8].• [9749989, 9749069989; 8].• [9755327, 9468755327; 8].• [9785903, 9795785903; 8].

• [9785903, 9795785903; 8].• [9789797, 9789797279; 8].• [9804911, 9863604911; 8].• [9827437, 9921827437; 8].• [9829447, 9829876447; 8].• [9832723, 9939832723; 8].• [9832723, 9939832723; 8].

• [9847577, 9845917577; 8].• [9869467, 9869467423; 8].• [9878731, 3429878731; 8].• [9893803, 9893852203; 8].• [9896147, 9896291147; 8].• [9905383, 9993605383; 8].• [9997937, 9971497937; 8].• [9999053, 9999282053; 8].

29


4.4 Four Digits RepetitionsBelow are few examples, with simplified notation.

• [2341, 22343341; 8] I 2 3412 2341 3412 2341 2341 3412 2341 2341 2341 3412 2341 2341 2341 2341 3412 2341 2341 2341 2341 2341 3412 2341 2341 2341 2341 2341 2341 3412 2341 2341 2341 2341 2341 2341 2341 341.

• [2341, 23419507; 8] I 23412341 95072341 9507 95072341 9507 9507 95072341 9507 9507 9507 95072341 9507 9507 9507 9507 95072341 9507 9507 9507 9507 9507 95072341 9507 9507 9507 9507 9507 9507 9507.


• [6551, 10296551; 8].• [9689, 97854689; 8].

• [15233, 152321033; 8].• [15233, 152321033; 8].• [33577, 126033577; 8].• [39779, 356289779; 8].• [43487, 434874389; 8].

• [53993, 539941163; 8].• [54163, 541689823; 8].• [56359, 563585089; 8].• [58631, 599818631; 8].

• [59239, 513449239; 8].• [67961, 679367561; 8].• [93187, 971523187; 8].• [97169, 971683109; 8].

• [133337, 1448533337; 8].• [163351, 1633621051; 8].• [168449, 1688745449; 8].• [173977, 1761053977; 8].• [177337, 1773391717; 8].

• [181297, 1815982297; 8].• [185299, 1130585299; 8].• [189713, 1897489313; 8].• [201337, 2013706237; 8].• [210323, 2105262323; 8].

• [233347, 2333455867; 8].• [234007, 2346642007; 8].• [245411, 2454608411; 8].• [245411, 2454608411; 8].• [247249, 2472493927; 8].

• [255133, 2554962133; 8].• [256981, 2566798981; 8].• [267131, 2679993131; 8].• [271429, 2714237389; 8].• [272039, 2720399447; 8].

• [279553, 2795536759; 8].• [284341, 2126384341; 8].• [285707, 1287285707; 8].• [299701, 2531399701; 8].• [301627, 3077731627; 8].

• [306253, 1416306253; 8].• [315037, 3589215037; 8].• [317693, 3176946533; 8].• [324217, 3245988217; 8].• [340027, 8043340027; 8].

30


• [360653, 3635040653; 8].• [374389, 3743899519; 8].• [374389, 3743899519; 8].• [379007, 3790067027; 8].• [380377, 3813080377; 8].

• [380377, 3813080377; 8].• [381949, 1065381949; 8].• [388057, 3636988057; 8].• [405089, 4052538089; 8].• [410173, 4174010173; 8].

• [410173, 4174010173; 8].• [422479, 4224715069; 8].• [422711, 9144422711; 8].• [435427, 4354296847; 8].• [463741, 3171463741; 8].

• [471619, 4791381619; 8].• [485029, 4850441329; 8].• [516209, 1260516209; 8].• [535943, 5359323443; 8].• [540079, 5409657079; 8].

• [558979, 6249558979; 8].• [567143, 5635257143; 8].• [577931, 5161777931; 8].• [600931, 6009337231; 8].• [600931, 6009337231; 8].

• [603487, 6034863427; 8].• [608897, 6035768897; 8].• [611707, 6427511707; 8].• [618707, 6844818707; 8].• [621871, 6241461871; 8].

• [627271, 6286197271; 8].• [643373, 6431683373; 8].• [643373, 6431683373; 8].• [646771, 6466246771; 8].• [646771, 6466246771; 8].• [646771, 6466246771; 8].

• [654553, 6545514973; 8].• [707939, 7099997939; 8].• [720899, 7202490899; 8].• [721199, 7999921199; 8].• [733793, 7848433793; 8].

• [766321, 7552066321; 8].• [812213, 8122148603; 8].• [818603, 8246618603; 8].• [820319, 8203110569; 8].• [820997, 8209042997; 8].

• [831739, 4407831739; 8].• [847853, 4062847853; 8].• [851041, 7161851041; 8].• [853703, 8537068073; 8].• [856469, 4722856469; 8].

• [869137, 7062869137; 8].• [898921, 8989492321; 8].• [922367, 9971122367; 8].• [947027, 9942347027; 8].

• [950611, 9506189131; 8].• [954977, 9571344977; 8].• [984931, 9847164931; 8].• [984931, 9847164931; 8].

4.5 Five Digits RepetitionsBelow are few examples, with simplified notation.

• [4597, 751924597; 8] I 459775192 459775192 75192 459775192 75192 75192 459775192 75192 75192 75192 459775192 75192 75192 75192 75192 459775192 75192 75192 75192 75192 75192 459775192 75192 75192 75192 75192 75192 75192 4597.

31


• [5081, 171215081; 8] I 508117121 508117121 17121 508117121 17121 17121 508117121 17121 17121 17121 508117121 17121 17121 17121 17121 508117121 17121 17121 17121 17121 17121 508117121 17121 17121 17121 17121 17121 17121 5081.


• [5443, 381125443; 8].• [5507, 550748819; 8].• [7027, 727573027; 8].• [8161, 409358161; 8].

• [14669, 7991114669; 8].• [14741, 1474269911; 8].• [15551, 1555527371; 8].• [16229, 1941496229; 8].

• [21283, 2128356427; 8].• [23087, 2385896087; 8].• [25919, 4040425919; 8].• [27851, 2785525481; 8].• [31883, 3252631883; 8].

• [31883, 3252631883; 8].• [33179, 3317564669; 8].• [34301, 3573814301; 8].• [38149, 3230108149; 8].

• [39317, 3931123277; 8].• [49783, 4978152853; 8].• [79319, 7931977511; 8].• [87931, 2877087931; 8].• [98717, 9871759943; 8].

• [105491, 77028105491; 8].• [107197, 15385607197; 8].• [109873, 10926472873; 8].• [113011, 11376620011; 8].• [117529, 11759802529; 8].

• [117529, 11759802529; 8].• [119569, 11956513369; 8].• [119569, 11956513369; 8].• [122827, 11083922827; 8].• [127301, 14824927301; 8].

• [132953, 13557402953; 8].• [135929, 13462185929; 8].• [141157, 14114776657; 8].• [144847, 13089444847; 8].• [156467, 15646248257; 8].

• [160387, 24708160387; 8].• [169523, 14206669523; 8].• [174347, 17434309127; 8].• [176383, 30900176383; 8].• [179533, 34209179533; 8].

• [184859, 18485195219; 8].• [185429, 12307085429; 8].• [191047, 19195523047; 8].• [199637, 19698199637; 8].• [199637, 19698199637; 8].

• [199637, 19698199637; 8].• [202841, 20187382841; 8].• [222329, 22275999329; 8].• [226549, 22654386319; 8].• [228337, 22835058637; 8].

• [231799, 24487431799; 8].• [233837, 23383918007; 8].• [235513, 23832925513; 8].• [246937, 24554706937; 8].• [248683, 98175248683; 8].

• [251117, 25150250117; 8].• [265483, 26543144083; 8].• [266281, 26628435811; 8].• [273149, 27318427649; 8].• [274237, 28952674237; 8].

• [283487, 28346571587; 8].• [297971, 29642007971; 8].• [298993, 29899103593; 8].• [298993, 29899103593; 8].• [300787, 37768800787; 8].

• [313553, 31358376653; 8].• [316097, 31680349097; 8].• [340339, 32404240339; 8].• [353117, 76602353117; 8].• [354551, 35455141751; 8].

• [354551, 35455141751; 8].• [354551, 35455141751; 8].• [357347, 35733979847; 8].• [358747, 35810431747; 8].• [360907, 31338660907; 8].

32


• [367547, 36114907547; 8].• [378713, 37876870313; 8].• [385661, 38566953671; 8].• [388259, 38822724359; 8].• [395137, 39513127207; 8].

• [407059, 41376707059; 8].• [408803, 45937908803; 8].• [408809, 40545618809; 8].• [410461, 41049769861; 8].• [410519, 41051475749; 8].

• [411563, 41799351563; 8].• [417509, 41750108849; 8].• [424573, 42677254573; 8].• [428557, 42810449557; 8].• [436673, 43667724743; 8].

• [453961, 45399618961; 8].• [453961, 45399618961; 8].• [454141, 49396654141; 8].• [454159, 45415664839; 8].• [460477, 46012372477; 8].

• [461059, 46179116059; 8].• [464081, 43340264081; 8].• [484201, 47594884201; 8].• [486907, 48691946707; 8].• [499801, 49982803501; 8].

• [501889, 86730501889; 8].• [513257, 55628913257; 8].• [516689, 51245136689; 8].• [517091, 52040617091; 8].• [519151, 51915323491; 8].

• [527333, 99015527333; 8].• [532823, 53567092823; 8].• [536729, 53675530229; 8].• [540961, 56547240961; 8].• [544199, 54419740799; 8].

• [544199, 54419740799; 8].• [557801, 70455557801; 8].• [566413, 56719466413; 8].• [566413, 56719466413; 8].• [584027, 58409570927; 8].

• [584557, 55103684557; 8].• [585779, 58579859279; 8].• [593869, 59386639999; 8].• [594959, 55345194959; 8].• [595117, 48285595117; 8].

• [595181, 59514951281; 8].• [600689, 60882960689; 8].• [605993, 60513941993; 8].• [626239, 62623991371; 8].• [638629, 21084638629; 8].

• [647081, 64750787081; 8].• [647081, 64750787081; 8].• [648079, 64808513179; 8].• [656951, 65634605951; 8].• [662083, 66208962493; 8].

• [663031, 66344970031; 8].• [668947, 91182668947; 8].• [669401, 44001669401; 8].• [676829, 69382576829; 8].• [677561, 54087677561; 8].

• [691153, 69120388153; 8].• [697937, 95019697937; 8].• [699053, 69312249053; 8].• [699073, 69342459073; 8].• [712601, 71217112601; 8].

• [712601, 71217112601; 8].• [712601, 71217112601; 8].• [723161, 72661353161; 8].• [730399, 73015015399; 8].• [731251, 73260311251; 8].

• [735871, 72149835871; 8].• [744451, 74392344451; 8].• [744451, 74392344451; 8].• [744941, 74499570941; 8].• [744941, 74499570941; 8].

• [754121, 68271754121; 8].• [775037, 67587775037; 8].• [782393, 77959082393; 8].• [803629, 80556743629; 8].• [828923, 82892988203; 8].

• [841849, 84184829419; 8].• [864107, 86410503227; 8].• [865829, 85776265829; 8].• [880603, 72315880603; 8].• [881171, 88255691171; 8].

• [883739, 88372036439; 8].• [886867, 88686789781; 8].• [925621, 92591479621; 8].• [928679, 92867319359; 8].• [940087, 94008858907; 8].

• [959597, 99473759597; 8].• [962309, 96310532309; 8].• [963901, 96396324001; 8].• [966313, 96638266513; 8].• [971029, 97767971029; 8].

• [971029, 97767971029; 8].• [971029, 97767971029; 8].• [971177, 97117630217; 8].• [977203, 97729587103; 8].• [989873, 13179989873; 8].

33


5 Patterns with Strings of Length 7In this section, we shall give examples of patterns with different digits repetitions for primes length 7. This hasbe divided according digits in the subsections below.

5.1 Single Digit RepetitionsFirst example given above can be represented two different ways. See below:

• [31, 331; 7] I 313 313 3 313 3 3 313 3 3 3 313 3 3 3 3 313 3 3 3 3 3 31.

• [31, 331; 7] I 313 3 13 3 3 313 3 3 3 313 3 3 3 3 313 3 3 3 3 3 313 3 3 3 3 3 3 31.

• [3557, 35507; 7] I 3557355 0 7355 0 0 7355 0 0 0 7355 0 0 0 0 7355 0 0 0 0 0 7355 0 0 0 0 0 0 7.

• [4549, 34549; 7] I 45493 45493 3 45493 3 3 45493 3 3 3 45493 3 3 3 3 45493 3 3 3 3 3 4549.

Below are more examples with reduced notations:

• [16561, 163561; 7].• [20071, 200713; 7].• [26431, 264331; 7].• [26431, 264331; 7].• [36563, 365063; 7].

• [37013, 337013; 7].• [37013, 337013; 7].• [37967, 379679; 7].• [40949, 409499; 7].• [40949, 409499; 7].

• [45317, 453317; 7].• [45317, 453317; 7].• [71287, 712807; 7].• [91811, 931811; 7].

• [106957, 1036957; 7].• [118843, 1189843; 7].• [185681, 1865681; 7].• [194977, 1949707; 7].• [195863, 1959863; 7].

• [201437, 2091437; 7].• [241441, 2431441; 7].• [278881, 2788831; 7].• [299311, 2993131; 7].• [352327, 3523327; 7].

• [352327, 3523327; 7].• [374837, 3748307; 7].• [401173, 4011373; 7].• [403363, 4036363; 7].• [414397, 4143697; 7].

• [471841, 4718419; 7].• [492047, 4920407; 7].• [559667, 6559667; 7].• [575129, 5751209; 7].• [589181, 5899181; 7].

• [589181, 5899181; 7].• [666119, 6660119; 7].• [699073, 6999073; 7].• [699073, 6999073; 7].• [699073, 6999073; 7].• [722377, 7223707; 7].

• [747259, 7472359; 7].• [815599, 8155099; 7].• [828523, 8285293; 7].• [853057, 8539057; 7].• [855719, 8557169; 7].• [952771, 9562771; 7].

• [1015073, 10165073; 7].• [1023019, 10230169; 7].• [1038623, 10338623; 7].• [1038623, 10338623; 7].• [1045367, 10495367; 7].

• [1078043, 10780463; 7].• [1095691, 10956931; 7].• [1208447, 19208447; 7].• [1217131, 12171319; 7].• [1217651, 12170651; 7].

• [1273687, 19273687; 7].• [1362103, 13621303; 7].• [1367059, 13673059; 7].• [1396987, 16396987; 7].• [1397311, 13973119; 7].

34


• [1434803, 14348093; 7].• [1438793, 14387939; 7].• [1446971, 14466971; 7].• [1446971, 14466971; 7].• [1455227, 14565227; 7].

• [1459597, 14595937; 7].• [1499963, 14099963; 7].• [1503613, 15036139; 7].• [1546399, 15406399; 7].• [1564457, 15664457; 7].

• [1564457, 15664457; 7].• [1589561, 15895631; 7].• [1623827, 16263827; 7].• [1654817, 16548317; 7].• [1685419, 16854139; 7].

• [1709261, 17092601; 7].• [1718369, 17198369; 7].• [1728691, 17286091; 7].• [1745333, 17435333; 7].• [1751377, 17501377; 7].

• [1809079, 18090379; 7].• [1827589, 31827589; 7].• [1918303, 19183093; 7].• [1993349, 19939349; 7].• [2055497, 23055497; 7].

• [2085151, 20851951; 7].• [2088421, 20884219; 7].• [2113249, 21136249; 7].• [2127721, 21277621; 7].• [2177297, 92177297; 7].

• [2234789, 22394789; 7].• [2249207, 22492073; 7].• [2251729, 22501729; 7].• [2261297, 92261297; 7].• [2274101, 22749101; 7].

• [2303057, 23063057; 7].• [2307187, 23097187; 7].• [2324129, 23241269; 7].• [2379023, 23790203; 7].• [2417027, 24173027; 7].

• [2437637, 23437637; 7].• [2471471, 24714713; 7].• [2692169, 26092169; 7].• [2715211, 27152911; 7].• [2722861, 27220861; 7].

• [2739859, 29739859; 7].• [2741083, 27941083; 7].• [2767361, 27697361; 7].• [2824879, 28234879; 7].• [2847839, 32847839; 7].

• [2894561, 28945061; 7].• [2940793, 62940793; 7].• [2949763, 29490763; 7].• [3017129, 30317129; 7].• [3082309, 30382309; 7].

• [3105857, 31905857; 7].• [3109697, 31096907; 7].• [3167207, 31672007; 7].• [3167207, 31672007; 7].• [3168757, 31608757; 7].

• [3222337, 32223937; 7].• [3245467, 32454607; 7].• [3265573, 32659573; 7].• [3280003, 32800039; 7].• [3306587, 39306587; 7].

• [3327371, 33270371; 7].• [3352277, 33592277; 7].• [3365617, 93365617; 7].• [3376627, 33976627; 7].• [3391433, 33919433; 7].

• [3481391, 39481391; 7].• [3492607, 34926079; 7].• [3525419, 35254319; 7].• [3527453, 35274539; 7].• [3606583, 33606583; 7].

• [3606583, 33606583; 7].• [3660721, 36607201; 7].• [3679183, 36791683; 7].• [3714397, 37314397; 7].• [3778081, 37798081; 7].

• [3778793, 37378793; 7].• [3784057, 37684057; 7].• [3788423, 37884023; 7].• [3837289, 38372869; 7].• [3853711, 38053711; 7].

• [3959209, 39659209; 7].• [3986531, 39686531; 7].• [3999487, 39699487; 7].• [4037333, 40373339; 7].• [4069847, 40669847; 7].

• [4069847, 40669847; 7].• [4209859, 94209859; 7].• [4221677, 42291677; 7].• [4337881, 43337881; 7].• [4337881, 43337881; 7].

• [4337881, 43337881; 7].• [4358539, 43958539; 7].• [4418581, 44185831; 7].• [4434421, 49434421; 7].• [4437737, 44377937; 7].

• [4480807, 44800807; 7].• [4480807, 44800807; 7].• [4490231, 44902331; 7].• [4490231, 44902331; 7].• [4493273, 44932793; 7].

• [4703683, 47036683; 7].• [4703683, 47036683; 7].• [4813579, 48136579; 7].• [4835333, 48356333; 7].• [4855237, 64855237; 7].

• [4871833, 48731833; 7].• [4886899, 48686899; 7].• [4928321, 49283021; 7].• [5021143, 56021143; 7].• [5046851, 50469851; 7].

• [5133239, 51332399; 7].• [5133239, 51332399; 7].• [5173409, 53173409; 7].• [5208967, 56208967; 7].• [5283167, 52083167; 7].

• [5286023, 52860263; 7].• [5397409, 53974909; 7].• [5485093, 54835093; 7].• [5497301, 54907301; 7].• [5633867, 35633867; 7].

• [5651593, 53651593; 7].• [5685833, 56085833; 7].• [5707607, 57067607; 7].• [5777887, 57778807; 7].• [5777917, 50777917; 7].

• [5793673, 57963673; 7].• [5860109, 58060109; 7].• [5864687, 95864687; 7].• [5900831, 59009831; 7].• [5941049, 65941049; 7].

• [5945161, 59451631; 7].• [6055541, 66055541; 7].• [6055541, 66055541; 7].• [6115667, 61153667; 7].• [6156739, 61567339; 7].• [6156739, 61567339; 7].• [6170977, 61709707; 7].• [6214357, 62143657; 7].• [6314279, 63149279; 7].• [6338117, 63381107; 7].

35


• [6347707, 66347707; 7].• [6347707, 66347707; 7].• [6349691, 63496991; 7].• [6349691, 63496991; 7].• [6522539, 65225393; 7].• [6528121, 65328121; 7].• [6562753, 65627563; 7].• [6563287, 65632387; 7].• [6616543, 69616543; 7].• [6646699, 69646699; 7].

• [6662041, 66620401; 7].• [6693641, 66963641; 7].• [6703087, 67603087; 7].• [6721843, 67218643; 7].• [6742709, 36742709; 7].• [6774821, 67674821; 7].• [7067491, 70674931; 7].• [7119227, 70119227; 7].• [7173239, 71732369; 7].• [7187087, 71867087; 7].

• [7191581, 70191581; 7].• [7319867, 73196867; 7].• [7396019, 70396019; 7].• [7409387, 74093807; 7].• [7448047, 73448047; 7].• [7456237, 73456237; 7].• [7457179, 74571769; 7].• [7573691, 75736091; 7].• [7578817, 75798817; 7].• [7640887, 76460887; 7].

• [7795343, 77935343; 7].• [7823029, 70823029; 7].• [7850543, 78506543; 7].• [7892461, 76892461; 7].• [7926241, 79926241; 7].• [7926241, 79926241; 7].• [7926847, 79326847; 7].• [7973327, 79733627; 7].• [8029069, 80269069; 7].• [8099219, 80909219; 7].

• [8221261, 82212661; 7].• [8221261, 82212661; 7].• [8234123, 82364123; 7].• [8235947, 82359647; 7].• [8275459, 98275459; 7].• [8282441, 82832441; 7].• [8306279, 83062079; 7].• [8510287, 85102873; 7].• [8571439, 85371439; 7].• [8616281, 68616281; 7].

• [8689661, 86896631; 7].• [8726633, 87266339; 7].• [8792101, 87962101; 7].• [8855503, 88055503; 7].• [8864147, 88641407; 7].• [8976787, 89767987; 7].• [9024679, 90240679; 7].• [9027877, 90927877; 7].• [9068861, 93068861; 7].• [9089719, 90089719; 7].

• [9089719, 90089719; 7].• [9102047, 91023047; 7].• [9151537, 93151537; 7].• [9185311, 93185311; 7].• [9188407, 91884007; 7].• [9188407, 91884007; 7].• [9254351, 92540351; 7].• [9305717, 93305717; 7].• [9305717, 93305717; 7].• [9308161, 93080161; 7].

• [9451019, 94501019; 7].• [9466447, 94664407; 7].• [9546431, 95464361; 7].• [9547273, 95472763; 7].• [9581023, 95681023; 7].• [9682997, 69682997; 7].• [9711881, 97119881; 7].• [9871843, 98718493; 7].• [9961607, 93961607; 7].

5.2 Double Digits RepetitionsBelow are some examples of representations:

• [1531, 121531; 7] I 153112 153112 12 153112 12 12 153112 12 12 12 153112 12 12 12 12 153112 12 12 12 12 12 1531.

• [1531, 121531; 7]. I 15311 21 5311 21 21 5311 21 21 21 5311 21 21 21 21 5311 21 21 21 21 21 5311 21 21 21 21 21 21 531.

The other examples are given in reduced form.

• [3733, 373327; 7].• [5737, 185737; 7].• [7691, 769751; 7].• [7727, 776627; 7].• [8543, 860543; 7].• [9403, 940399; 7].

• [14143, 1414423; 7].• [20071, 2390071; 7].• [20287, 6620287; 7].• [20333, 2420333; 7].• [20333, 2420333; 7].• [22067, 6322067; 7].

• [28871, 2218871; 7].• [35327, 3533627; 7].• [40739, 4073159; 7].• [43331, 4375331; 7].• [44059, 4404559; 7].• [45599, 4485599; 7].

• [47977, 4797607; 7].• [49117, 6649117; 7].• [55673, 5536673; 7].• [57493, 5743393; 7].• [84211, 8422411; 7].• [84653, 8465381; 7].

36


• [85903, 7285903; 7].• [85903, 8590357; 7].• [86509, 8650849; 7].• [95467, 9596467; 7].• [95713, 9725713; 7].

• [101737, 10541737; 7].• [106823, 10682993; 7].• [111697, 11169157; 7].• [123377, 13323377; 7].• [125063, 45125063; 7].

• [125441, 12305441; 7].• [126019, 12630019; 7].• [127781, 12776081; 7].• [128153, 12899153; 7].• [136511, 19636511; 7].

• [138101, 13878101; 7].• [138101, 13878101; 7].• [140627, 14330627; 7].• [141689, 14145689; 7].

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• [307481, 30748241; 7].• [319399, 31459399; 7].• [320057, 32018057; 7].• [321889, 32871889; 7].

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• [581863, 58127863; 7].• [586067, 21586067; 7].• [599023, 59981023; 7].• [605039, 60503819; 7].

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• [729587, 72958997; 7].• [730399, 73540399; 7].• [741491, 74149211; 7].• [747283, 74728903; 7].

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• [761377, 78761377; 7].• [763031, 76302731; 7].• [765329, 76599329; 7].• [772573, 77233573; 7].

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• [817837, 81783907; 7].• [832631, 48832631; 7].• [841069, 84106249; 7].• [847423, 89047423; 7].

• [852427, 85452427; 7].• [852427, 85452427; 7].• [858167, 85813967; 7].• [867689, 86765489; 7].

37


• [869257, 86925187; 7].• [871103, 87110339; 7].• [873727, 87633727; 7].• [877109, 45877109; 7].

• [879743, 87974573; 7].• [899681, 90899681; 7].• [906427, 90648127; 7].• [911527, 91781527; 7].

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38


• [1803079, 184803079; 7].• [1830349, 183099349; 7].• [1838719, 183998719; 7].• [1845913, 145845913; 7].

• [1846139, 184426139; 7].• [1870651, 184270651; 7].• [1883177, 188313077; 7].• [1890719, 189071669; 7].

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• [2537081, 253708181; 7].• [2537081, 253708181; 7].• [2537081, 253708181; 7].• [2540071, 236540071; 7].

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• [2927047, 292709647; 7].• [2934623, 293634623; 7].• [2934623, 297834623; 7].• [2934623, 293634623; 7].

39


• [2959949, 299059949; 7].• [2968181, 296398181; 7].• [2968549, 296398549; 7].• [2988901, 298890661; 7].

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• [4373147, 437372147; 7].• [4381549, 431281549; 7].• [4386457, 434586457; 7].• [4438999, 724438999; 7].

40


• [4481713, 448176613; 7].• [4485743, 448665743; 7].• [4488331, 448833631; 7].• [4488331, 448833631; 7].

• [4492157, 449215337; 7].• [4499009, 449219009; 7].• [4516189, 472516189; 7].• [4521311, 874521311; 7].

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• [5102599, 510182599; 7].• [5129129, 512912369; 7].• [5194093, 519334093; 7].• [5247163, 965247163; 7].

• [5254063, 525406663; 7].• [5254063, 525406663; 7].• [5266979, 526697909; 7].• [5271557, 527172557; 7].

• [5294651, 125294651; 7].• [5304577, 530664577; 7].• [5328233, 532368233; 7].• [5330107, 533019607; 7].

• [5334751, 535434751; 7].• [5345569, 534995569; 7].• [5350931, 535054931; 7].• [5364269, 536544269; 7].

• [5378267, 537826337; 7].• [5435519, 543587519; 7].• [5442641, 544572641; 7].• [5442893, 544289243; 7].

• [5458231, 549658231; 7].• [5510977, 551097697; 7].• [5517077, 551637077; 7].• [5531161, 553116241; 7].

• [5551807, 125551807; 7].• [5568817, 545568817; 7].• [5568817, 545568817; 7].• [5580517, 558053017; 7].

• [5590271, 365590271; 7].• [5596187, 559633187; 7].• [5606707, 584606707; 7].• [5616469, 561643669; 7].

• [5642909, 564290981; 7].• [5650363, 568750363; 7].• [5692669, 875692669; 7].• [5693881, 551693881; 7].

• [5694761, 569476811; 7].• [5714453, 577514453; 7].• [5714857, 527714857; 7].• [5717563, 530717563; 7].

• [5717963, 554717963; 7].• [5733929, 905733929; 7].• [5754779, 575544779; 7].• [5755733, 573355733; 7].

• [5764501, 485764501; 7].• [5813333, 581333783; 7].• [5816971, 533816971; 7].• [5857463, 551857463; 7].

• [5879239, 587921839; 7].• [5880463, 588066463; 7].• [5896459, 587896459; 7].• [5896459, 587896459; 7].

• [5899211, 589921151; 7].• [5899211, 589921151; 7].• [5915401, 591536401; 7].• [5922173, 845922173; 7].

• [5933537, 593333537; 7].• [5933537, 593333537; 7].• [5933537, 593333537; 7].• [5961647, 597861647; 7].

• [6000073, 600005773; 7].• [6007123, 876007123; 7].• [6025753, 602572153; 7].• [6029041, 602901841; 7].

• [6033857, 603324857; 7].• [6043153, 936043153; 7].• [6057193, 605739193; 7].• [6096469, 605796469; 7].

• [6118711, 648118711; 7].• [6134981, 306134981; 7].• [6138299, 663138299; 7].• [6150283, 615078283; 7].

• [6168359, 616638359; 7].• [6175577, 617563577; 7].• [6183623, 618243623; 7].• [6183857, 618333857; 7].

41


• [6183857, 618333857; 7].• [6197941, 619794139; 7].• [6230831, 623083691; 7].• [6235751, 666235751; 7].

• [6235751, 666235751; 7].• [6250421, 625044521; 7].• [6250961, 699250961; 7].• [6267257, 626187257; 7].

• [6290611, 629990611; 7].• [6290611, 629990611; 7].• [6291611, 246291611; 7].• [6295687, 657295687; 7].

• [6306731, 632106731; 7].• [6309547, 996309547; 7].• [6312071, 631208471; 7].• [6314197, 660314197; 7].

• [6317503, 615317503; 7].• [6335449, 633335449; 7].• [6335449, 633335449; 7].• [6335449, 633335449; 7].

• [6342499, 634246099; 7].• [6393857, 996393857; 7].• [6411571, 641157169; 7].• [6431713, 643171873; 7].

• [6437957, 643792757; 7].• [6461821, 646451821; 7].• [6481927, 669481927; 7].• [6538201, 653820781; 7].

• [6552901, 336552901; 7].• [6553621, 655362751; 7].• [6556439, 651256439; 7].• [6570731, 657990731; 7].

• [6583531, 659083531; 7].• [6586147, 658657147; 7].• [6598817, 660598817; 7].• [6642379, 664290379; 7].

• [6688657, 786688657; 7].• [6695947, 669592747; 7].• [6699881, 669999881; 7].• [6699881, 669999881; 7].

• [6699881, 669999881; 7].• [6700777, 670077739; 7].• [6703687, 186703687; 7].• [6717401, 671607401; 7].

• [6731311, 699731311; 7].• [6750577, 669750577; 7].• [6775721, 677533721; 7].• [6797503, 679217503; 7].

• [6799613, 679981613; 7].• [6800719, 680077219; 7].• [6807923, 686607923; 7].• [6816973, 681876973; 7].

• [6827687, 682712687; 7].• [6828667, 682833667; 7].• [6832037, 576832037; 7].• [6851261, 685128461; 7].

• [6905669, 186905669; 7].• [6906817, 692406817; 7].• [6939589, 693958459; 7].• [6951467, 695146997; 7].

• [6955537, 695555137; 7].• [6970693, 698770693; 7].• [6977107, 697733107; 7].• [6999361, 633999361; 7].

• [7027567, 702157567; 7].• [7030613, 577030613; 7].• [7049453, 704945663; 7].• [7055107, 705519307; 7].

• [7055369, 705537869; 7].• [7088461, 703088461; 7].• [7094551, 709455931; 7].• [7128727, 739128727; 7].

• [7131613, 719031613; 7].• [7143217, 733143217; 7].• [7152619, 715233619; 7].• [7157987, 733157987; 7].

• [7181017, 790181017; 7].• [7193579, 713693579; 7].• [7204621, 720465721; 7].• [7211861, 517211861; 7].

• [7212353, 367212353; 7].• [7213391, 721339331; 7].• [7214843, 727214843; 7].• [7214843, 727214843; 7].

• [7214843, 727214843; 7].• [7239577, 723957757; 7].• [7239577, 723957757; 7].• [7251523, 725172523; 7].

• [7273817, 725473817; 7].• [7298441, 751298441; 7].• [7307009, 732707009; 7].• [7321123, 732112783; 7].

• [7322891, 790322891; 7].• [7324661, 487324661; 7].• [7350257, 735025667; 7].• [7357891, 732757891; 7].

• [7358479, 731558479; 7].• [7360883, 736021883; 7].• [7361839, 736151839; 7].• [7361839, 736151839; 7].

• [7390667, 247390667; 7].• [7404451, 740445163; 7].• [7416109, 743616109; 7].• [7446487, 744648187; 7].

• [7446487, 744648187; 7].• [7447357, 744735793; 7].• [7448479, 744847669; 7].• [7451417, 745331417; 7].

• [7467511, 746757211; 7].• [7478479, 427478479; 7].• [7497029, 739497029; 7].• [7506217, 750621337; 7].

• [7520659, 718520659; 7].• [7521893, 367521893; 7].• [7534069, 753406219; 7].• [7538093, 753869093; 7].

• [7548803, 754758803; 7].• [7549067, 775549067; 7].• [7583137, 758314537; 7].• [7585393, 758554393; 7].

• [7599413, 759975413; 7].• [7626719, 799626719; 7].• [7647491, 764724491; 7].• [7670809, 767048809; 7].

• [7681543, 877681543; 7].• [7732651, 773261851; 7].• [7773401, 457773401; 7].• [7784783, 778904783; 7].

• [7790639, 766790639; 7].• [7816541, 781996541; 7].• [7837919, 783397919; 7].• [7852613, 697852613; 7].

• [7866191, 786613691; 7].• [7869403, 637869403; 7].• [7872793, 787962793; 7].• [7898551, 187898551; 7].

• [7900117, 337900117; 7].• [7965007, 791565007; 7].• [7987081, 798337081; 7].• [7996867, 799688767; 7].

42


• [7998013, 799338013; 7].• [8019541, 801954157; 7].• [8024927, 887024927; 7].• [8035003, 854035003; 7].

• [8046187, 804618757; 7].• [8046187, 804618757; 7].• [8054023, 805403323; 7].• [8063221, 806322991; 7].

• [8065033, 805765033; 7].• [8084737, 899084737; 7].• [8103971, 812703971; 7].• [8125309, 338125309; 7].

• [8129567, 812999567; 7].• [8129567, 812999567; 7].• [8131741, 813174163; 7].• [8148757, 814884757; 7].

• [8157311, 815731841; 7].• [8160179, 817260179; 7].• [8165401, 816540601; 7].• [8187623, 815487623; 7].

• [8205073, 820335073; 7].• [8217323, 548217323; 7].• [8229103, 158229103; 7].• [8243681, 824368421; 7].

• [8252539, 825602539; 7].• [8271943, 638271943; 7].• [8279683, 368279683; 7].• [8279899, 827489899; 7].

• [8287847, 824587847; 7].• [8289551, 308289551; 7].• [8310397, 834810397; 7].• [8319121, 831913021; 7].

• [8342119, 832442119; 7].• [8347993, 899347993; 7].• [8355419, 428355419; 7].• [8361329, 638361329; 7].

• [8389709, 838970399; 7].• [8391641, 668391641; 7].• [8402683, 840263383; 7].• [8408929, 815408929; 7].

• [8412673, 863412673; 7].• [8465669, 884465669; 7].• [8467777, 884467777; 7].• [8472713, 844872713; 7].

• [8482549, 875482549; 7].• [8500859, 850075859; 7].• [8518427, 851833427; 7].• [8560303, 188560303; 7].

• [8572253, 857152253; 7].• [8632223, 938632223; 7].• [8635559, 863559959; 7].• [8638933, 863893153; 7].

• [8664959, 869664959; 7].• [8664959, 869664959; 7].• [8693197, 248693197; 7].• [8708143, 758708143; 7].

• [8725567, 866725567; 7].• [8755853, 488755853; 7].• [8782211, 458782211; 7].• [8794921, 879454921; 7].

• [8794921, 879454921; 7].• [8799023, 879579023; 7].• [8810939, 896810939; 7].• [8824883, 882488333; 7].

• [8824883, 882488333; 7].• [8830907, 883078907; 7].• [8831981, 883192481; 7].• [8841337, 889941337; 7].

• [8847589, 338847589; 7].• [8850001, 885001801; 7].• [8852663, 824852663; 7].• [8857063, 885706813; 7].

• [8859709, 848859709; 7].• [8859709, 848859709; 7].• [8867549, 886367549; 7].• [8867549, 886367549; 7].

• [8871113, 887172113; 7].• [8879813, 854879813; 7].• [8885603, 218885603; 7].• [8887897, 888789751; 7].

• [8940097, 894021097; 7].• [8967457, 896277457; 7].• [8982769, 898332769; 7].• [8985863, 898566863; 7].

• [9009509, 904809509; 7].• [9014623, 957014623; 7].• [9027167, 902715167; 7].• [9027167, 902715167; 7].

• [9056651, 905666351; 7].• [9067189, 960067189; 7].• [9068183, 904868183; 7].• [9071203, 907751203; 7].

• [9078037, 907803937; 7].• [9078037, 907803937; 7].• [9086837, 945086837; 7].• [9094697, 909469937; 7].

• [9103909, 639103909; 7].• [9130321, 913032361; 7].• [9144413, 914446613; 7].• [9156209, 915628409; 7].

• [9161809, 914261809; 7].• [9165257, 999165257; 7].• [9165257, 999165257; 7].• [9175871, 519175871; 7].

• [9177001, 917790001; 7].• [9190411, 993190411; 7].• [9225217, 922521277; 7].• [9231647, 923199647; 7].

• [9237581, 925137581; 7].• [9250123, 925015123; 7].• [9250123, 925015123; 7].• [9262577, 925762577; 7].

• [9263669, 926813669; 7].• [9273527, 927993527; 7].• [9283739, 579283739; 7].• [9299161, 929930161; 7].

• [9300409, 930330409; 7].• [9301429, 963301429; 7].• [9306589, 930658957; 7].• [9310349, 931810349; 7].

• [9310349, 931810349; 7].• [9317681, 931847681; 7].• [9324433, 932436433; 7].• [9331667, 933931667; 7].

• [9331667, 933931667; 7].• [9341713, 789341713; 7].• [9355573, 935557351; 7].• [9359813, 935993813; 7].

• [9360697, 936600697; 7].• [9374153, 966374153; 7].• [9392359, 939692359; 7].• [9392359, 939692359; 7].

• [9393701, 339393701; 7].• [9423761, 942330761; 7].• [9425527, 942552277; 7].• [9433267, 943366267; 7].

• [9454997, 945394997; 7].• [9471947, 947331947; 7].• [9477337, 219477337; 7].• [9552313, 955122313; 7].

43


• [9566047, 933566047; 7].• [9575717, 957571787; 7].• [9575717, 957571787; 7].• [9637697, 963764297; 7].• [9666091, 966609151; 7].• [9666091, 966609151; 7].• [9682627, 968262637; 7].• [9687299, 968748299; 7].• [9700601, 970750601; 7].• [9716107, 915716107; 7].• [9717749, 971766749; 7].

• [9726029, 729726029; 7].• [9743011, 945743011; 7].• [9770279, 977960279; 7].• [9770737, 942770737; 7].• [9787867, 978786877; 7].• [9799373, 973099373; 7].• [9812447, 981244751; 7].• [9824119, 982484119; 7].• [9824119, 982484119; 7].• [9848131, 984815131; 7].• [9848131, 984815131; 7].

• [9849319, 986049319; 7].• [9860197, 986015197; 7].• [9860197, 986015197; 7].• [9865993, 986599993; 7].• [9865993, 986599993; 7].• [9865993, 986599993; 7].• [9874691, 987469751; 7].• [9880327, 988240327; 7].• [9885763, 988518763; 7].• [9893539, 989213539; 7].• [9910499, 189910499; 7].

• [9920653, 992093653; 7].• [9926159, 996326159; 7].• [9945941, 994965941; 7].• [9954667, 995814667; 7].• [9956753, 995651753; 7].• [9970291, 997029961; 7].• [9981739, 994581739; 7].• [9982831, 998283631; 7].• [9982831, 998283631; 7].• [9996247, 999156247; 7].

5.3 Triple Digits RepetitionsBelow are some examples of representations:

• [467, 570467; 7] I 467570 467570 570 467570 570 570 467570 570 570 570 467570 570 570 570 570 467570 570 570 570 570 570 467.

• [857, 812057; 7]. I 8571 120 578 120 120 578 120 120 120 578 120 120 120 120 578 120 120 120 120 120 578 120 120 120 120 120 120 57.


• [1229, 1222889; 7].• [1327, 1387327; 7].• [1327, 1387327; 7].• [1361, 1279361; 7].• [2153, 1982153; 7].• [2281, 2285401; 7].• [3769, 3138769; 7].• [4931, 4965131; 7].

• [5807, 5893907; 7].• [5927, 5957027; 7].• [6257, 6216257; 7].• [6257, 6216257; 7].• [6257, 6216257; 7].• [6763, 5226763; 7].• [7019, 7011479; 7].• [7187, 7182767; 7].

• [7517, 8827517; 7].• [7867, 7696867; 7].• [8117, 8196017; 7].• [8821, 8836621; 7].• [9091, 9096781; 7].• [9103, 9444103; 7].• [9967, 9967327; 7].• [9967, 9967327; 7].

• [10193, 10193357; 7].• [10831, 13630831; 7].• [11491, 11498731; 7].• [11681, 11685731; 7].• [11941, 14951941; 7].• [12589, 12582289; 7].• [12589, 12582289; 7].• [13267, 17113267; 7].

• [13267, 17113267; 7].• [13567, 13948567; 7].• [14107, 18824107; 7].• [14431, 14467231; 7].• [14537, 14597537; 7].• [14537, 14597537; 7].• [14563, 13844563; 7].• [14767, 14769967; 7].

• [14767, 14769967; 7].• [16193, 16197113; 7].• [16363, 16366213; 7].• [16553, 16556153; 7].• [16553, 16556153; 7].• [16561, 16933561; 7].• [17443, 17439343; 7].• [17491, 82817491; 7].

• [17881, 17836381; 7].• [18041, 51318041; 7].• [18233, 96618233; 7].• [18253, 66918253; 7].• [18587, 28518587; 7].• [18797, 18731597; 7].• [19319, 19365119; 7].• [19801, 51619801; 7].

• [20441, 20446031; 7].• [21839, 21336839; 7].• [22669, 27382669; 7].• [23099, 23040299; 7].• [23117, 23117477; 7].• [23117, 23117477; 7].• [23531, 23596031; 7].• [23857, 23987857; 7].

44


• [24631, 24691831; 7].• [25243, 25846243; 7].• [27739, 27736369; 7].• [28387, 28312987; 7].• [28817, 28817639; 7].• [30323, 30246323; 7].• [31687, 31051687; 7].• [31699, 31691899; 7].

• [31699, 31691899; 7].• [32531, 36332531; 7].• [32531, 36332531; 7].• [33749, 33744239; 7].• [34361, 34363661; 7].• [34361, 34363661; 7].• [35141, 35630141; 7].• [35159, 35194559; 7].

• [36653, 36408653; 7].• [38971, 38972221; 7].• [39857, 39854447; 7].• [41117, 44411117; 7].• [43411, 43462411; 7].• [43411, 43462411; 7].• [43541, 43544951; 7].• [46327, 43036327; 7].

• [46559, 46180559; 7].• [46703, 48346703; 7].• [46703, 48346703; 7].• [47441, 47420441; 7].• [47441, 47420441; 7].• [47857, 47824657; 7].• [48757, 51348757; 7].• [49333, 49320133; 7].

• [49939, 49917739; 7].• [51109, 59391109; 7].• [51817, 51857917; 7].• [51949, 99951949; 7].• [55889, 61855889; 7].• [55997, 55991477; 7].• [56629, 56622799; 7].• [56957, 56952527; 7].

• [57143, 57141353; 7].• [57259, 57255769; 7].• [57559, 57559669; 7].• [57559, 57559669; 7].• [58043, 81658043; 7].• [58043, 58038143; 7].• [58567, 58168567; 7].• [58567, 58168567; 7].

• [58789, 58772089; 7].• [58889, 28858889; 7].• [59239, 97559239; 7].• [63097, 63372097; 7].• [63149, 63163649; 7].• [64271, 69904271; 7].• [64871, 69334871; 7].• [65357, 65310857; 7].

• [65479, 65479693; 7].• [65983, 65816983; 7].• [66491, 66477791; 7].• [67043, 62197043; 7].• [68171, 24968171; 7].• [68171, 61178171; 7].• [69119, 69158519; 7].• [69593, 69759593; 7].

• [69593, 69759593; 7].• [69929, 36369929; 7].• [70991, 70935191; 7].• [72421, 72426091; 7].• [72959, 72966659; 7].• [73079, 41473079; 7].• [73907, 20473907; 7].• [74717, 22274717; 7].

• [75787, 75787969; 7].• [76603, 76180603; 7].• [78467, 76638467; 7].• [79349, 80179349; 7].• [81899, 81872099; 7].• [84053, 22584053; 7].• [85819, 85813579; 7].• [86077, 82196077; 7].

• [86171, 86175251; 7].• [86263, 86114263; 7].• [87223, 52887223; 7].• [87739, 87739591; 7].• [89561, 89567351; 7].• [93089, 93084029; 7].• [93719, 93797819; 7].• [93893, 93003893; 7].

• [94439, 94387439; 7].• [94543, 94521043; 7].• [95189, 95969189; 7].• [95869, 98495869; 7].• [95869, 98495869; 7].• [96643, 91926643; 7].• [97187, 98347187; 7].• [99409, 99297409; 7].

• [100417, 197800417; 7].• [102397, 102397153; 7].• [103091, 103098041; 7].• [103979, 103973609; 7].• [103993, 159703993; 7].• [104789, 104789117; 7].• [105319, 105756319; 7].• [105967, 105967117; 7].

• [105967, 105967117; 7].• [107687, 107687777; 7].• [107687, 107687777; 7].• [108637, 107178637; 7].• [109201, 105109201; 7].• [109201, 105109201; 7].• [109807, 109806097; 7].• [110711, 110711987; 7].

• [110927, 110654927; 7].• [111271, 504111271; 7].• [112759, 116572759; 7].• [113809, 113853409; 7].• [114781, 114781999; 7].• [115223, 115223897; 7].• [116903, 113336903; 7].• [117937, 118407937; 7].

• [118373, 118378433; 7].• [118913, 116278913; 7].• [122833, 172922833; 7].• [123077, 123073787; 7].• [124669, 130024669; 7].• [125509, 123575509; 7].• [127541, 177727541; 7].• [130631, 130662131; 7].

• [132113, 132119783; 7].• [132529, 127632529; 7].• [133709, 148033709; 7].• [134581, 134582641; 7].• [136163, 136168673; 7].• [137771, 137778191; 7].• [137947, 137941597; 7].• [138113, 138987113; 7].

• [138209, 116538209; 7].• [139759, 825139759; 7].• [140381, 140594381; 7].• [140417, 447140417; 7].• [140603, 194540603; 7].• [140863, 140582863; 7].• [140909, 140909969; 7].• [140909, 140909969; 7].

• [141413, 111141413; 7].• [141413, 111141413; 7].• [142841, 142844231; 7].• [143551, 143234551; 7].• [145007, 145007747; 7].• [145007, 145007747; 7].• [145547, 116545547; 7].• [147997, 147999727; 7].

• [149711, 149713721; 7].• [150011, 150015251; 7].• [150769, 528150769; 7].• [152123, 152123927; 7].• [152239, 152237689; 7].• [152753, 152738153; 7].• [153001, 133053001; 7].• [153443, 153870443; 7].

45


• [153743, 585153743; 7].• [154081, 154095181; 7].• [155501, 155120501; 7].• [155627, 134855627; 7].• [157103, 159397103; 7].• [157303, 155797303; 7].• [157649, 240157649; 7].• [158617, 158619667; 7].

• [159157, 474159157; 7].• [160033, 160033969; 7].• [160603, 161980603; 7].• [161039, 161828039; 7].• [162527, 136062527; 7].• [163477, 387163477; 7].• [164147, 164182847; 7].• [164363, 164345963; 7].

• [165469, 165437269; 7].• [166669, 816166669; 7].• [168601, 168663601; 7].• [168601, 168663601; 7].• [168899, 789168899; 7].• [169249, 169242979; 7].• [170029, 780170029; 7].• [170669, 170612369; 7].

• [171481, 177831481; 7].• [171713, 171713501; 7].• [175873, 175813873; 7].• [175873, 175813873; 7].• [176611, 121076611; 7].• [176899, 648176899; 7].• [178643, 178660943; 7].• [179261, 179266331; 7].

• [179429, 179421839; 7].• [179497, 179464897; 7].• [179749, 179742859; 7].• [180749, 180744899; 7].• [181919, 141181919; 7].• [181919, 141181919; 7].• [182027, 182028827; 7].• [182027, 182028827; 7].

• [182129, 187742129; 7].• [183509, 183503399; 7].• [183683, 175383683; 7].• [184369, 184362469; 7].• [184369, 184362469; 7].• [187009, 187720009; 7].• [187559, 187539059; 7].• [187963, 187837963; 7].

• [187963, 187837963; 7].• [188473, 188615473; 7].• [188653, 188658073; 7].• [188827, 124088827; 7].• [188833, 182738833; 7].• [189421, 189427801; 7].• [190657, 714190657; 7].• [192047, 192438047; 7].

• [193327, 459193327; 7].• [194707, 408194707; 7].• [196769, 193516769; 7].• [197221, 160697221; 7].• [198301, 198305671; 7].• [198929, 194058929; 7].• [199081, 199615081; 7].• [199483, 199393483; 7].

• [199967, 199966667; 7].• [199967, 199966667; 7].• [200579, 200633579; 7].• [202493, 202789493; 7].• [205721, 399205721; 7].• [207523, 207423523; 7].• [208609, 208697509; 7].• [209621, 201599621; 7].

• [209971, 214109971; 7].• [212281, 257612281; 7].• [212353, 212353789; 7].• [212627, 212834627; 7].• [213281, 213213281; 7].• [213281, 213213281; 7].• [213281, 213213281; 7].• [213281, 213213281; 7].

• [213281, 213213281; 7].• [214381, 231214381; 7].• [214381, 231214381; 7].• [215359, 215165359; 7].• [215359, 215165359; 7].• [217429, 217499929; 7].• [217489, 217489711; 7].• [218137, 218131657; 7].

• [218531, 123218531; 7].• [219311, 618219311; 7].• [220243, 220285243; 7].• [220243, 220285243; 7].• [220771, 220771471; 7].• [220771, 220771471; 7].• [220771, 220771471; 7].• [221489, 234521489; 7].

• [221659, 942221659; 7].• [221873, 221877053; 7].• [222109, 492222109; 7].• [223217, 295123217; 7].• [224579, 224207579; 7].• [224717, 251024717; 7].• [225163, 287025163; 7].• [225961, 225156961; 7].

• [226141, 271726141; 7].• [227299, 227342299; 7].• [227797, 227742997; 7].• [227993, 282227993; 7].• [227993, 282227993; 7].• [228341, 228338741; 7].• [230393, 235130393; 7].• [230393, 235130393; 7].

• [230683, 255230683; 7].• [230683, 255230683; 7].• [230941, 233870941; 7].• [231559, 263631559; 7].• [231589, 231589723; 7].• [231871, 231876871; 7].• [231871, 231876871; 7].• [231871, 231876871; 7].

• [232189, 238552189; 7].• [232861, 266032861; 7].• [234457, 234774457; 7].• [234457, 234774457; 7].• [236813, 236177813; 7].• [237301, 390237301; 7].• [238039, 114238039; 7].• [238967, 801238967; 7].

• [243917, 861243917; 7].• [244303, 244307293; 7].• [244339, 244813339; 7].• [244411, 244415671; 7].• [244721, 244728431; 7].• [245711, 245719091; 7].• [246713, 246828713; 7].• [246787, 246657787; 7].

• [246811, 246432811; 7].• [247879, 268147879; 7].• [248041, 248091541; 7].• [249449, 672249449; 7].• [251099, 251082599; 7].• [251533, 271151533; 7].• [251879, 257561879; 7].• [253109, 253696109; 7].

• [254803, 254642803; 7].• [255023, 255023399; 7].• [256771, 236056771; 7].• [257711, 257737211; 7].• [258743, 251328743; 7].• [259201, 295759201; 7].• [259379, 259780379; 7].• [260089, 213860089; 7].

46


• [260467, 260463277; 7].• [261223, 261227503; 7].• [261241, 375261241; 7].• [261463, 261771463; 7].• [261463, 261771463; 7].• [263267, 624263267; 7].• [265313, 265321613; 7].• [265861, 265840261; 7].

• [268069, 153268069; 7].• [269749, 269312749; 7].• [270973, 270975763; 7].• [272249, 290972249; 7].• [272717, 272714297; 7].• [272981, 272981231; 7].• [272981, 272981231; 7].• [273367, 273362347; 7].

• [273473, 277203473; 7].• [275167, 275169847; 7].• [275699, 275576699; 7].• [276239, 298476239; 7].• [276953, 271446953; 7].• [277787, 277939787; 7].• [277897, 277893397; 7].• [277897, 277893397; 7].

• [278807, 278854907; 7].• [279991, 279994861; 7].• [283793, 283794113; 7].• [284447, 284446187; 7].• [284561, 284366561; 7].• [286459, 276286459; 7].• [286459, 276286459; 7].• [288257, 288253307; 7].

• [288989, 272388989; 7].• [291359, 291358289; 7].• [291539, 291927539; 7].• [293129, 292793129; 7].• [293129, 292793129; 7].• [293681, 293683121; 7].• [294731, 294840731; 7].• [297019, 296037019; 7].

• [297533, 297552533; 7].• [297533, 297552533; 7].• [298451, 298519451; 7].• [298679, 217498679; 7].• [298759, 294928759; 7].• [299359, 299363959; 7].• [300799, 300606799; 7].• [303767, 303718367; 7].

• [304253, 304250153; 7].• [304849, 304810249; 7].• [305441, 305448701; 7].• [306193, 306453193; 7].• [306301, 308376301; 7].• [306563, 306561623; 7].• [307511, 309397511; 7].• [307543, 321607543; 7].

• [309403, 309407863; 7].• [309541, 810309541; 7].• [310049, 310041089; 7].• [313727, 316963727; 7].• [315251, 315281951; 7].• [317983, 317981953; 7].• [318377, 318387677; 7].• [319001, 319015901; 7].

• [325201, 311125201; 7].• [325891, 609325891; 7].• [328043, 867328043; 7].• [328067, 326128067; 7].• [328067, 326128067; 7].• [330839, 330849839; 7].• [330839, 330849839; 7].• [331241, 471331241; 7].

• [331777, 359431777; 7].• [332573, 332573603; 7].• [332573, 332573603; 7].• [333023, 333058223; 7].• [333337, 333333757; 7].• [334183, 334155883; 7].• [335897, 335898887; 7].• [336689, 336688109; 7].

• [337091, 337079291; 7].• [340201, 346420201; 7].• [343627, 186343627; 7].• [345067, 345069607; 7].• [345263, 344085263; 7].• [346139, 346137089; 7].• [346217, 346215377; 7].• [349291, 349996291; 7].

• [350191, 352700191; 7].• [350443, 350413543; 7].• [351259, 351861259; 7].• [351259, 351861259; 7].• [351293, 915351293; 7].• [353329, 353329321; 7].• [353869, 300353869; 7].• [354869, 354594869; 7].

• [354869, 354594869; 7].• [355519, 355513819; 7].• [355519, 355513819; 7].• [355549, 627355549; 7].• [358159, 358183759; 7].• [360779, 360777479; 7].• [360779, 360777479; 7].• [361271, 364981271; 7].

• [361687, 361933687; 7].• [361927, 368401927; 7].• [362137, 362120137; 7].• [362137, 362120137; 7].• [362191, 364712191; 7].• [362707, 362701567; 7].• [362723, 391562723; 7].• [365747, 364445747; 7].

• [366313, 350766313; 7].• [366329, 366329723; 7].• [367637, 376567637; 7].• [369071, 369071663; 7].• [369361, 366369361; 7].• [369361, 366369361; 7].• [369361, 366369361; 7].• [369637, 963369637; 7].

• [369673, 369462673; 7].• [371177, 371294177; 7].• [371719, 825371719; 7].• [372271, 372711271; 7].• [374783, 374105783; 7].• [375283, 375261883; 7].• [377491, 377552491; 7].• [378101, 379968101; 7].

• [382171, 382176841; 7].• [384089, 385944089; 7].• [384247, 384244327; 7].• [384317, 516384317; 7].• [385141, 385124041; 7].• [387151, 807387151; 7].• [390101, 390106181; 7].• [390877, 391320877; 7].

• [393137, 397203137; 7].• [393637, 393366637; 7].• [396091, 396621091; 7].• [396247, 380496247; 7].• [396953, 396940253; 7].• [397549, 397544359; 7].• [397799, 372397799; 7].• [397799, 372397799; 7].

• [399983, 399923483; 7].• [400093, 400020193; 7].• [405437, 405437741; 7].• [407657, 425507657; 7].• [408437, 408399437; 7].• [408643, 493008643; 7].• [409027, 409327027; 7].• [411833, 411836483; 7].

47


• [413071, 413071819; 7].• [414083, 414082013; 7].• [414467, 306414467; 7].• [414793, 414747493; 7].• [415379, 415343879; 7].• [418603, 418936603; 7].• [419561, 419562581; 7].• [420551, 420555131; 7].

• [421699, 421699561; 7].• [423847, 421893847; 7].• [426311, 426987311; 7].• [428401, 428456401; 7].• [428401, 428456401; 7].• [429487, 429483337; 7].• [429659, 429627659; 7].• [429659, 429627659; 7].

• [431863, 431828863; 7].• [431863, 431828863; 7].• [434977, 434973967; 7].• [435817, 436275817; 7].• [436483, 446536483; 7].• [436801, 436856401; 7].• [436963, 434686963; 7].• [436999, 438976999; 7].

• [437201, 437129201; 7].• [440303, 480740303; 7].• [440893, 732440893; 7].• [441631, 666441631; 7].• [442807, 459742807; 7].• [445183, 445840183; 7].• [445463, 445477763; 7].• [446087, 446099987; 7].

• [447101, 447181901; 7].• [448561, 448562671; 7].• [448867, 448459867; 7].• [450533, 450517433; 7].• [452957, 453932957; 7].• [455159, 455159297; 7].• [455167, 615455167; 7].• [456991, 852456991; 7].

• [457013, 828457013; 7].• [457987, 491857987; 7].• [458069, 458069813; 7].• [458921, 451928921; 7].• [458993, 459728993; 7].• [459113, 453429113; 7].• [459313, 459313279; 7].• [460969, 460965979; 7].

• [462263, 462741263; 7].• [464281, 294464281; 7].• [465077, 465078227; 7].• [465541, 465541309; 7].• [466913, 418366913; 7].• [467737, 464237737; 7].• [470219, 470213099; 7].• [471781, 471724681; 7].

• [472063, 478462063; 7].• [473173, 473176153; 7].• [473351, 473276351; 7].• [474311, 478494311; 7].• [474437, 696474437; 7].• [475159, 475112959; 7].• [479821, 915479821; 7].• [479881, 479881021; 7].

• [480563, 873480563; 7].• [483397, 488793397; 7].• [484193, 486694193; 7].• [486323, 488466323; 7].• [486949, 484836949; 7].• [488227, 417188227; 7].• [489653, 489654563; 7].• [489733, 485829733; 7].

• [490579, 490858579; 7].• [491159, 491136959; 7].• [492047, 492171047; 7].• [493133, 495853133; 7].• [493313, 493313423; 7].• [493313, 493313423; 7].• [495041, 418995041; 7].• [495569, 495566849; 7].

• [495787, 491565787; 7].• [497677, 180497677; 7].• [497711, 444797711; 7].• [499391, 493489391; 7].• [499693, 564499693; 7].• [499969, 499963249; 7].• [500953, 500913253; 7].• [501157, 501157159; 7].

• [504103, 541104103; 7].• [504299, 504213299; 7].• [504299, 504213299; 7].• [504527, 504523727; 7].• [504527, 504523727; 7].• [505481, 505624481; 7].• [506531, 471506531; 7].• [508957, 508954597; 7].

• [509417, 509416997; 7].• [509737, 505949737; 7].• [511541, 514531541; 7].• [512591, 512521991; 7].• [518159, 512378159; 7].• [518233, 597818233; 7].• [518863, 581318863; 7].• [519919, 225519919; 7].

• [519997, 561819997; 7].• [521767, 551321767; 7].• [524219, 524211059; 7].• [524801, 524531801; 7].• [525583, 524895583; 7].• [526387, 526383997; 7].• [526397, 526393997; 7].• [526397, 526393997; 7].

• [526997, 559426997; 7].• [526997, 526300997; 7].• [527053, 527771053; 7].• [527753, 527753789; 7].• [529577, 525349577; 7].• [529933, 529931863; 7].• [530183, 533330183; 7].• [530183, 533330183; 7].

• [530339, 530338559; 7].• [532907, 532980407; 7].• [533723, 429533723; 7].• [533837, 533837321; 7].• [536293, 537626293; 7].• [537773, 537773711; 7].• [542219, 542213129; 7].• [543463, 543463171; 7].

• [543787, 543324787; 7].• [544009, 544732009; 7].• [546019, 546011419; 7].• [546019, 546011419; 7].• [546683, 547776683; 7].• [547369, 547379869; 7].• [547787, 554947787; 7].• [548059, 548059363; 7].

• [550447, 550273447; 7].• [551483, 552071483; 7].• [556271, 147556271; 7].• [557027, 557067827; 7].• [558253, 582858253; 7].• [559831, 554089831; 7].• [560149, 560435149; 7].• [561461, 561933461; 7].

• [566759, 566754299; 7].• [567871, 567786871; 7].• [568657, 568426657; 7].• [569507, 234569507; 7].• [570403, 571590403; 7].• [575593, 575599933; 7].• [578467, 578454367; 7].• [578609, 518978609; 7].

48


• [579133, 579136213; 7].• [579197, 579124697; 7].• [581687, 559781687; 7].• [587219, 587726219; 7].• [587737, 587195737; 7].• [588433, 511488433; 7].• [590753, 599030753; 7].• [590839, 596510839; 7].

• [591457, 592821457; 7].• [592517, 592536617; 7].• [595139, 537895139; 7].• [596293, 596291863; 7].• [598981, 598985461; 7].• [599657, 596039657; 7].• [600293, 600243893; 7].• [601231, 609601231; 7].

• [601231, 609601231; 7].• [602821, 602825401; 7].• [602891, 675302891; 7].• [605177, 601985177; 7].• [607517, 607111517; 7].• [608843, 608843663; 7].• [608843, 608843663; 7].• [610843, 618100843; 7].

• [612173, 612173279; 7].• [613297, 613212697; 7].• [613651, 613626751; 7].• [613981, 613925881; 7].• [614657, 614659967; 7].• [617051, 613907051; 7].• [617761, 615407761; 7].• [617887, 611087887; 7].

• [620657, 620675057; 7].• [620831, 291620831; 7].• [621451, 621738451; 7].• [624329, 753624329; 7].• [625997, 625942697; 7].• [627541, 627548431; 7].• [629023, 621779023; 7].• [631859, 631853249; 7].

• [633079, 867633079; 7].• [634531, 693634531; 7].• [634531, 693634531; 7].• [634709, 634703639; 7].• [635729, 624635729; 7].• [635729, 624635729; 7].• [635813, 635872313; 7].• [635983, 632675983; 7].

• [636313, 648336313; 7].• [637201, 637206151; 7].• [640229, 258640229; 7].• [640993, 630940993; 7].• [641057, 641666057; 7].• [644057, 644076557; 7].• [644363, 972644363; 7].• [644837, 642584837; 7].

• [645077, 645084977; 7].• [645727, 645793927; 7].• [647987, 644087987; 7].• [649151, 697249151; 7].• [650549, 650549639; 7].• [650549, 650549639; 7].• [650779, 650715679; 7].• [651191, 611751191; 7].

• [651869, 651117869; 7].• [654727, 654729877; 7].• [656291, 652286291; 7].• [658547, 658520747; 7].• [659591, 366659591; 7].• [660733, 660736633; 7].• [660733, 660736633; 7].

• [661097, 661096817; 7].• [661603, 661631203; 7].• [663127, 663549127; 7].• [664211, 664858211; 7].• [666607, 669816607; 7].• [666751, 666754381; 7].• [671141, 689471141; 7].• [671161, 671303161; 7].• [672169, 672169297; 7].

• [672557, 678342557; 7].• [673109, 309673109; 7].• [675067, 675453067; 7].• [675109, 675178909; 7].• [675841, 675889441; 7].• [679039, 637579039; 7].• [682141, 682143151; 7].• [682337, 682375037; 7].

• [684769, 684741469; 7].• [686449, 682496449; 7].• [688453, 688984453; 7].• [688573, 669088573; 7].• [688741, 631288741; 7].• [689867, 689863877; 7].• [689957, 688379957; 7].• [692149, 698942149; 7].

• [692221, 692540221; 7].• [692779, 699842779; 7].• [693127, 693199027; 7].• [694079, 694034879; 7].• [694091, 694055891; 7].• [694783, 694984783; 7].• [694783, 694984783; 7].• [694951, 683494951; 7].

• [696323, 696951323; 7].• [696827, 696823937; 7].• [697217, 474697217; 7].• [698083, 228698083; 7].• [698891, 698824391; 7].• [699379, 667599379; 7].• [700897, 700893607; 7].• [701443, 701441593; 7].

• [701579, 303701579; 7].• [708647, 706188647; 7].• [712289, 787912289; 7].• [713407, 716963407; 7].• [714443, 714621443; 7].• [714509, 713454509; 7].• [715571, 719635571; 7].• [718381, 719728381; 7].

• [718559, 718294559; 7].• [719057, 719056727; 7].• [720023, 720028613; 7].• [720887, 720869687; 7].• [721381, 721315381; 7].• [721381, 721315381; 7].• [722369, 722345069; 7].• [725359, 725352829; 7].

• [725603, 558725603; 7].• [726521, 727686521; 7].• [726953, 726979553; 7].• [728027, 453728027; 7].• [729403, 729468703; 7].• [730619, 730615829; 7].• [732449, 732449747; 7].• [733939, 733648939; 7].

• [735173, 123735173; 7].• [736721, 737866721; 7].• [740143, 740342143; 7].• [741599, 741252599; 7].• [742697, 742697471; 7].• [743849, 743845469; 7].• [744911, 741204911; 7].• [745273, 828745273; 7].

• [745951, 486745951; 7].• [747811, 747811459; 7].• [748021, 745258021; 7].• [748057, 743068057; 7].• [748211, 748280411; 7].• [752639, 759482639; 7].• [754289, 754255289; 7].• [754289, 754255289; 7].

49


• [754711, 754963711; 7].• [756443, 756448493; 7].• [756463, 757236463; 7].• [758341, 756638341; 7].• [759673, 231759673; 7].• [760723, 763120723; 7].• [761381, 761386331; 7].• [762373, 882762373; 7].

• [764131, 764108131; 7].• [766877, 618766877; 7].• [768059, 768513059; 7].• [768319, 767508319; 7].• [769459, 769684459; 7].• [770359, 770359963; 7].• [770641, 778310641; 7].• [771473, 771843473; 7].

• [772073, 772073831; 7].• [772789, 772246789; 7].• [772859, 772884059; 7].• [773029, 351773029; 7].• [773951, 738173951; 7].• [774127, 774160027; 7].• [774131, 774131429; 7].• [777391, 777372691; 7].

• [777463, 777465373; 7].• [779837, 776489837; 7].• [780613, 501780613; 7].• [783557, 783530057; 7].• [785579, 330785579; 7].• [785773, 782315773; 7].• [785777, 785390777; 7].• [786017, 786019907; 7].

• [786901, 180786901; 7].• [786949, 786943219; 7].• [787217, 705787217; 7].• [787879, 787880179; 7].• [788971, 788970271; 7].• [790861, 790204861; 7].• [791573, 791553773; 7].• [793841, 793841309; 7].

• [794449, 794457049; 7].• [797567, 797568767; 7].• [797567, 797568767; 7].• [797977, 797606977; 7].• [800783, 803210783; 7].• [800801, 884300801; 7].• [805019, 859705019; 7].• [807197, 807195197; 7].

• [807197, 807195197; 7].• [807197, 807195197; 7].• [807409, 120807409; 7].• [808373, 808376813; 7].• [809821, 803489821; 7].• [810259, 810231259; 7].• [810259, 810231259; 7].• [810571, 810521671; 7].

• [811561, 818221561; 7].• [812717, 812717783; 7].• [813817, 814083817; 7].• [814043, 814098143; 7].• [815123, 857015123; 7].• [815471, 827015471; 7].• [817867, 817885867; 7].• [817867, 817885867; 7].

• [818353, 818330053; 7].• [819911, 888519911; 7].• [820429, 663820429; 7].• [821879, 821869379; 7].• [822161, 824232161; 7].• [823499, 823499891; 7].• [824081, 824327081; 7].• [825697, 825288697; 7].

• [825983, 823905983; 7].• [826019, 826062719; 7].• [827087, 585827087; 7].• [829967, 309829967; 7].• [832583, 234832583; 7].• [832933, 832934503; 7].• [835039, 810535039; 7].• [835249, 835903249; 7].

• [836159, 836194859; 7].• [838463, 838933463; 7].• [841751, 841750751; 7].• [841751, 841750751; 7].• [841931, 826141931; 7].• [842767, 842766037; 7].• [843443, 843467843; 7].• [844183, 844732183; 7].

• [844429, 844423819; 7].• [845459, 844895459; 7].• [847163, 847119263; 7].• [848941, 877448941; 7].• [849997, 849786997; 7].• [851449, 842051449; 7].• [851573, 851579123; 7].• [852143, 852159743; 7].

• [855271, 856185271; 7].• [856381, 856371181; 7].• [858427, 867558427; 7].• [860513, 846860513; 7].• [860513, 846860513; 7].• [863119, 863456119; 7].• [864509, 864378509; 7].• [865177, 357865177; 7].

• [865409, 865475609; 7].• [866231, 270866231; 7].• [871001, 879661001; 7].• [871061, 879601061; 7].• [871439, 171871439; 7].• [871513, 871771513; 7].• [871513, 871771513; 7].• [871513, 871771513; 7].

• [872107, 856472107; 7].• [874583, 874759583; 7].• [874681, 874963681; 7].• [875089, 875753089; 7].• [877117, 875197117; 7].• [877313, 872497313; 7].• [878443, 878449423; 7].• [879341, 879346751; 7].

• [880069, 880018969; 7].• [883237, 883765237; 7].• [886313, 886355213; 7].• [886973, 885646973; 7].• [890737, 890900737; 7].• [892321, 873892321; 7].• [892321, 873892321; 7].• [893449, 893442559; 7].

• [895529, 354895529; 7].• [896447, 896444537; 7].• [897163, 889497163; 7].• [897229, 897248029; 7].• [897629, 897698729; 7].• [898199, 892588199; 7].• [900019, 901440019; 7].• [902569, 902565319; 7].

• [904019, 516904019; 7].• [904823, 904246823; 7].• [906487, 906492487; 7].• [906487, 906492487; 7].• [906707, 906707573; 7].• [906881, 906222881; 7].• [907447, 906607447; 7].• [907663, 714907663; 7].

• [908137, 908145037; 7].• [908221, 432908221; 7].• [910421, 910318421; 7].• [910817, 910129817; 7].• [911011, 246911011; 7].• [911023, 950411023; 7].• [911839, 916991839; 7].• [915703, 915777103; 7].

50


• [916061, 408916061; 7].• [916073, 916297073; 7].• [916367, 916368647; 7].• [916571, 916515671; 7].• [919229, 919930229; 7].• [920197, 920124397; 7].• [920497, 920436997; 7].• [921001, 921001273; 7].• [924557, 924587957; 7].• [928621, 966928621; 7].• [928621, 966928621; 7].• [929311, 921479311; 7].• [932317, 150932317; 7].• [933923, 933925253; 7].• [934613, 934683713; 7].• [937049, 989437049; 7].

• [937777, 937731877; 7].• [939511, 939512011; 7].• [939511, 939512011; 7].• [939677, 936429677; 7].• [939853, 939851833; 7].• [940889, 940903889; 7].• [941179, 962741179; 7].• [945359, 255945359; 7].

• [946331, 942106331; 7].• [950357, 950312057; 7].• [950423, 950432123; 7].• [950867, 950887667; 7].• [953969, 916553969; 7].• [954011, 303954011; 7].• [954221, 738954221; 7].• [956929, 956924119; 7].• [957317, 959097317; 7].• [957977, 929757977; 7].• [959471, 959478491; 7].• [961783, 964741783; 7].• [962237, 962233247; 7].• [962617, 965912617; 7].• [963043, 177963043; 7].• [963643, 963641233; 7].

• [964351, 964352551; 7].• [964351, 964352551; 7].• [965411, 965633411; 7].• [966583, 966583183; 7].• [966583, 966583183; 7].• [966583, 966583183; 7].• [966677, 963966677; 7].• [966677, 963966677; 7].

• [966677, 963966677; 7].• [971153, 971192153; 7].• [971153, 971192153; 7].• [972431, 972431441; 7].• [972431, 972431441; 7].• [973151, 912373151; 7].• [974317, 975674317; 7].• [974317, 975674317; 7].• [974819, 786974819; 7].• [974819, 974850419; 7].• [974999, 974999999; 7].• [974999, 974999999; 7].• [974999, 974999999; 7].• [974999, 974999999; 7].• [976453, 976456783; 7].• [976853, 922876853; 7].

• [977567, 977565317; 7].• [977593, 977174593; 7].• [978713, 978717953; 7].• [979001, 979034801; 7].• [980069, 980084669; 7].• [980489, 986300489; 7].• [983699, 944183699; 7].• [985631, 985652831; 7].

• [986131, 993386131; 7].• [987739, 981087739; 7].• [987739, 981087739; 7].• [988909, 988318909; 7].• [988909, 988318909; 7].• [989321, 989396621; 7].• [991409, 969991409; 7].• [991409, 969991409; 7].

• [991531, 997981531; 7].• [991663, 991662883; 7].• [993107, 993103757; 7].• [993913, 997413913; 7].• [994183, 994186363; 7].• [995783, 999035783; 7].

• [996167, 996501167; 7].• [997699, 992917699; 7].• [998197, 998196547; 7].• [998381, 998984381; 7].• [999199, 132999199; 7].

5.4 Four Digits RepetitionsBelow are some examples of representations:

• [191, 2145191; 7] I 1912145 1912145 2145 1912145 2145 2145 1912145 2145 2145 2145 1912145 2145 2145 2145 2145 1912145 2145 2145 2145 2145 2145 191.

• [283, 2246683; 7]. I 2832 2466 832 2466 2466 832 2466 2466 2466 832 2466 2466 2466 2466 832 2466 2466 2466 2466 2466 832 2466 2466 2466 2466 2466 2466 83.


• [353, 3321653; 7].• [373, 8859373; 7].• [419, 4193663; 7].• [613, 6719413; 7].• [619, 6148489; 7].• [659, 7062659; 7].

• [1123, 11236777; 7].• [1427, 14894027; 7].• [1433, 14377043; 7].• [1597, 32401597; 7].• [1663, 16632373; 7].• [1663, 16632373; 7].

• [1721, 17238521; 7].• [1721, 17238521; 7].• [1733, 17314673; 7].• [1753, 17286253; 7].• [2621, 25955621; 7].• [2861, 28660031; 7].

• [2927, 29218757; 7].• [2953, 20552953; 7].• [3011, 31386011; 7].• [3041, 30363041; 7].• [3041, 30363041; 7].• [3259, 35592259; 7].

51


• [3851, 38562581; 7].• [4219, 42132439; 7].• [4243, 47839243; 7].• [4603, 46017553; 7].• [4733, 47363033; 7].• [4733, 47363033; 7].

• [4751, 47536121; 7].• [4943, 49432853; 7].• [4943, 49432853; 7].• [5857, 58574977; 7].• [5857, 58574977; 7].• [6277, 62773681; 7].

• [6397, 49566397; 7].• [6397, 63582697; 7].• [6521, 65652221; 7].• [6763, 67631281; 7].• [7351, 73518559; 7].• [7537, 22777537; 7].• [8609, 81335609; 7].• [9013, 90134053; 7].• [9013, 90134053; 7].• [9463, 94668883; 7].

• [10099, 420910099; 7].• [10501, 114700501; 7].• [11119, 116366119; 7].• [11719, 176171719; 7].• [11813, 158381813; 7].• [11827, 118391827; 7].• [11827, 118391827; 7].• [11827, 118391827; 7].• [12497, 124975649; 7].• [12689, 124833689; 7].

• [13177, 131913877; 7].• [13487, 269113487; 7].• [13907, 503113907; 7].• [14159, 149888159; 7].• [14159, 141537959; 7].• [14159, 141537959; 7].• [15233, 152418533; 7].• [15473, 154780673; 7].• [15473, 154780673; 7].• [15671, 190095671; 7].

• [15749, 111765749; 7].• [16091, 587416091; 7].• [17041, 281417041; 7].• [17093, 176765093; 7].• [18061, 180644281; 7].• [18131, 797418131; 7].• [18413, 184646213; 7].• [18517, 281118517; 7].• [18553, 185549653; 7].• [18553, 185549653; 7].

• [19301, 456919301; 7].• [19387, 193975087; 7].• [19427, 455419427; 7].• [19777, 192970777; 7].• [19973, 199735073; 7].• [19973, 199735073; 7].• [19973, 199735073; 7].• [19997, 199911617; 7].• [20161, 266330161; 7].• [20369, 203661839; 7].

• [20599, 205915999; 7].• [20599, 205915999; 7].• [20773, 205220773; 7].• [20773, 205220773; 7].• [21397, 208821397; 7].• [21739, 245721739; 7].• [21739, 245721739; 7].• [22147, 221451487; 7].• [22349, 223494881; 7].• [24023, 285774023; 7].

• [24049, 879324049; 7].• [24763, 192324763; 7].• [25969, 259665319; 7].• [26437, 958826437; 7].• [26839, 268381879; 7].• [26959, 269144059; 7].• [27893, 278942903; 7].• [28109, 281562209; 7].• [29339, 293317499; 7].• [29629, 226409629; 7].

• [29947, 701429947; 7].• [30137, 301953737; 7].• [30467, 336510467; 7].• [30803, 308045033; 7].• [31231, 501631231; 7].• [33599, 335918159; 7].• [34057, 344836057; 7].• [34171, 341659771; 7].• [34513, 345183313; 7].• [34513, 345183313; 7].

• [35311, 353387611; 7].• [35869, 358676629; 7].• [36007, 360106807; 7].• [36151, 361199551; 7].• [36857, 368517167; 7].• [36973, 393696973; 7].• [37057, 324097057; 7].• [37201, 372719401; 7].• [37811, 951337811; 7].• [37831, 378172231; 7].

• [37963, 379280263; 7].• [38873, 388757783; 7].• [39521, 395669921; 7].• [39623, 399789623; 7].• [39623, 399789623; 7].• [39979, 374199979; 7].• [40087, 276040087; 7].• [40277, 402777953; 7].• [40343, 403315943; 7].• [40813, 408194203; 7].

• [40849, 408451489; 7].• [41081, 410888771; 7].• [41491, 414187591; 7].• [41579, 411422579; 7].• [42281, 645342281; 7].• [42281, 422643281; 7].• [42281, 422643281; 7].• [42701, 478872701; 7].• [43487, 434871023; 7].• [44293, 846344293; 7].

• [44357, 442961357; 7].• [44587, 445888837; 7].• [45427, 454273843; 7].• [45497, 454978949; 7].• [46411, 464175511; 7].• [46411, 464175511; 7].• [46447, 464571847; 7].• [46457, 464552867; 7].• [46751, 463855751; 7].• [46771, 420136771; 7].

• [46853, 419626853; 7].• [47189, 757247189; 7].• [47513, 466997513; 7].• [49433, 494270333; 7].• [49823, 492253823; 7].• [49927, 950149927; 7].• [50051, 563030051; 7].• [50359, 515000359; 7].• [51283, 512842093; 7].• [51437, 519390437; 7].

• [51673, 516308773; 7].• [51829, 518285179; 7].• [52069, 455452069; 7].• [52583, 530722583; 7].• [52981, 575332981; 7].• [53117, 538229117; 7].• [53861, 538652201; 7].• [53899, 528983899; 7].• [54091, 540962581; 7].• [54101, 341454101; 7].

52


• [54367, 543657487; 7].• [54577, 838254577; 7].• [55061, 528445061; 7].• [55843, 558107743; 7].• [56167, 565727167; 7].• [56269, 924056269; 7].• [56401, 564412501; 7].• [56737, 567399457; 7].• [57389, 115557389; 7].• [57667, 576779767; 7].

• [58321, 571468321; 7].• [58979, 570628979; 7].• [60913, 215460913; 7].• [62171, 628031171; 7].• [63419, 634199003; 7].• [64451, 682744451; 7].• [64717, 647115187; 7].• [66089, 648876089; 7].• [66373, 879066373; 7].• [66383, 663816893; 7].

• [66403, 664364803; 7].• [66449, 664818449; 7].• [66449, 664818449; 7].• [66949, 669434629; 7].• [67043, 375367043; 7].• [67103, 671063513; 7].• [67589, 675863549; 7].• [67631, 676654331; 7].• [67759, 673972759; 7].• [68227, 645188227; 7].

• [68993, 689917763; 7].• [69001, 690921001; 7].• [69193, 691241893; 7].• [69763, 695706763; 7].• [70901, 701383901; 7].• [70979, 555970979; 7].• [71191, 184871191; 7].• [72101, 721490501; 7].• [72173, 728622173; 7].• [72173, 728622173; 7].

• [72467, 724637027; 7].• [72733, 727339939; 7].• [73459, 731239459; 7].• [73651, 870973651; 7].• [73727, 737212247; 7].• [73867, 731815867; 7].• [73961, 739494761; 7].• [74177, 741757847; 7].• [74189, 772034189; 7].• [74521, 763364521; 7].

• [74521, 745132021; 7].• [74779, 747798133; 7].• [74869, 748618399; 7].• [75013, 130875013; 7].• [75217, 752989717; 7].• [75223, 795345223; 7].• [75833, 751260833; 7].• [76471, 717766471; 7].• [80039, 857270039; 7].• [80429, 804400829; 7].

• [80599, 805973599; 7].• [80599, 805973599; 7].• [80599, 805973599; 7].• [80909, 809817809; 7].• [81637, 815277637; 7].• [82067, 820676279; 7].• [84199, 889704199; 7].• [84307, 846039307; 7].• [84809, 870624809; 7].• [85607, 802585607; 7].

• [86197, 861973039; 7].• [86767, 867647227; 7].• [86843, 903986843; 7].• [87049, 860067049; 7].• [87257, 877299257; 7].• [87323, 873280913; 7].• [88237, 882352297; 7].• [89491, 894954091; 7].• [89491, 894954091; 7].• [91297, 913879297; 7].

• [91499, 917644499; 7].• [92227, 970142227; 7].• [92753, 927429353; 7].• [92899, 926045899; 7].• [92927, 158792927; 7].• [93427, 939756427; 7].• [93971, 630393971; 7].• [94009, 948067009; 7].• [95219, 990515219; 7].• [95419, 307595419; 7].

• [95621, 782795621; 7].• [95773, 891095773; 7].• [95947, 161795947; 7].• [96043, 998406043; 7].• [96479, 184896479; 7].• [97423, 974274073; 7].• [97943, 979479233; 7].• [98041, 980412781; 7].• [98041, 980412781; 7].• [98411, 925718411; 7].

• [98869, 329798869; 7].• [99089, 990664889; 7].• [99149, 783699149; 7].• [99581, 788799581; 7].• [99611, 996315911; 7].• [100511, 1005337811; 7].• [100907, 1002559907; 7].• [101027, 1086821027; 7].• [101063, 1010625683; 7].• [101383, 1120301383; 7].

• [101483, 1014813203; 7].• [103979, 1650103979; 7].• [103979, 1650103979; 7].• [105107, 1370505107; 7].• [105449, 9084105449; 7].• [105817, 1284105817; 7].• [105817, 1284105817; 7].• [106619, 1066213019; 7].• [106907, 1069693307; 7].• [107089, 2157107089; 7].

• [107251, 1072519207; 7].• [107903, 1046477903; 7].• [107927, 1012787927; 7].• [108293, 1082993783; 7].• [108359, 1091418359; 7].• [109063, 8022109063; 7].• [109507, 1091362507; 7].• [110939, 2568110939; 7].• [110939, 1172090939; 7].• [111043, 1110343543; 7].

• [111611, 1117893611; 7].• [111773, 1835811773; 7].• [112163, 2394112163; 7].• [112967, 1112872967; 7].• [112997, 1122400997; 7].• [114043, 1147818043; 7].• [114089, 8073114089; 7].• [115861, 8040115861; 7].• [115883, 1158863933; 7].• [116047, 1160267347; 7].

• [116201, 1165019201; 7].• [117023, 1162467023; 7].• [117413, 1541517413; 7].• [117577, 1175939977; 7].• [117721, 1177412521; 7].• [117851, 1178405651; 7].• [118913, 1189142813; 7].• [118913, 1189142813; 7].• [119869, 1198610239; 7].• [120011, 8295120011; 7].

• [121259, 1212544709; 7].• [121291, 6549121291; 7].• [121637, 1227241637; 7].• [122387, 1264472387; 7].• [122489, 1224741989; 7].• [122509, 1225092283; 7].• [122609, 6810122609; 7].• [123143, 5124123143; 7].• [123209, 1232401409; 7].• [123911, 1239114263; 7].

53


• [124363, 8034124363; 7].• [125201, 1515425201; 7].• [125383, 1253651683; 7].• [125591, 1251089591; 7].• [126349, 1267770349; 7].• [126733, 1267337569; 7].• [127817, 1247827817; 7].• [127817, 1247827817; 7].• [128339, 1215818339; 7].• [128833, 9870128833; 7].

• [129737, 1299693737; 7].• [131839, 1362821839; 7].• [131899, 1311245899; 7].• [131969, 1384271969; 7].• [132469, 1324709569; 7].• [133261, 3450133261; 7].• [134333, 2442134333; 7].• [134399, 1397024399; 7].• [134951, 1360724951; 7].• [134989, 1349881969; 7].

• [136651, 1366591381; 7].• [137273, 1377335273; 7].• [137507, 1375082957; 7].• [138739, 1386204739; 7].• [139291, 1331779291; 7].• [139333, 1393319383; 7].• [139369, 1398286369; 7].• [139759, 1397541469; 7].• [140473, 1715840473; 7].• [140677, 6825140677; 7].

• [141671, 1433121671; 7].• [141803, 1418057033; 7].• [142369, 1423697281; 7].• [142699, 6852142699; 7].• [143573, 1164443573; 7].• [144773, 1447309373; 7].• [145121, 3255145121; 7].• [145771, 1917445771; 7].• [146941, 1469971141; 7].• [147299, 1270647299; 7].

• [149053, 1354649053; 7].• [149333, 1493337011; 7].• [150767, 1509567767; 7].• [150889, 1507089889; 7].• [151013, 1519408013; 7].• [151273, 1549321273; 7].• [152027, 1480052027; 7].• [152407, 1524077371; 7].• [152791, 1798352791; 7].• [153191, 1531947701; 7].

• [154501, 1545906001; 7].• [155663, 1551155663; 7].• [155663, 1551155663; 7].• [155663, 1551155663; 7].• [155663, 1551155663; 7].• [156011, 1560990911; 7].• [156269, 5316156269; 7].• [156601, 1566030541; 7].• [156799, 1567995211; 7].• [157051, 1570525141; 7].

• [157411, 1546027411; 7].• [157523, 1520317523; 7].• [157951, 1574938951; 7].• [160243, 1601980243; 7].• [161641, 1616473531; 7].• [161773, 1747061773; 7].• [162251, 1622561261; 7].• [162787, 1627590187; 7].• [163127, 3267163127; 7].• [163433, 1639948433; 7].

• [163859, 1639570859; 7].• [165833, 5334165833; 7].• [165857, 1686465857; 7].• [165857, 1686465857; 7].• [167159, 1671520709; 7].• [167269, 1679750269; 7].• [168533, 1685336831; 7].• [169009, 1656919009; 7].• [169019, 6129169019; 7].• [169777, 1681609777; 7].

• [169909, 1804369909; 7].• [169951, 1699544551; 7].• [169951, 1699544551; 7].• [170021, 1777970021; 7].• [170021, 1777970021; 7].• [170207, 1399970207; 7].• [171043, 8064171043; 7].• [171659, 1716783659; 7].• [171659, 1716783659; 7].• [172199, 6549172199; 7].

• [172597, 1725308797; 7].• [173039, 1766423039; 7].• [173807, 1887773807; 7].• [174091, 1748022091; 7].• [174929, 1712214929; 7].• [175649, 8823175649; 7].• [175837, 1758647437; 7].• [176213, 1762905713; 7].• [176921, 1761692921; 7].• [177091, 1773903091; 7].

• [178561, 1785607561; 7].• [178561, 1785607561; 7].• [178921, 1789838221; 7].• [179161, 1791670831; 7].• [179213, 1645679213; 7].• [179393, 2835179393; 7].• [180161, 1801149761; 7].• [180241, 1802172241; 7].• [180241, 1802172241; 7].• [180347, 2112180347; 7].

• [181081, 1249081081; 7].• [182029, 1820793529; 7].• [183289, 1870023289; 7].• [183377, 1888503377; 7].• [184489, 1465684489; 7].• [185569, 3630185569; 7].• [186037, 1869144037; 7].• [186947, 1869496457; 7].• [188137, 1339988137; 7].• [188941, 1326788941; 7].

• [190063, 1902394063; 7].• [191047, 1962971047; 7].• [191627, 1916221787; 7].• [191669, 1760891669; 7].• [191861, 1723091861; 7].• [193301, 1933046861; 7].• [193811, 8613193811; 7].• [194069, 1990844069; 7].• [194819, 1943234819; 7].• [194819, 1943234819; 7].

• [194933, 1264394933; 7].• [197383, 1973849953; 7].• [197947, 4719197947; 7].• [198197, 2865198197; 7].• [198937, 1981626937; 7].• [200191, 2001960991; 7].• [200191, 2001960991; 7].• [202129, 2027269129; 7].

• [202907, 2023387907; 7].• [203221, 2032363621; 7].• [203843, 2031176843; 7].• [203977, 2039220577; 7].• [204353, 2043585473; 7].• [204623, 2550804623; 7].• [205063, 2050639471; 7].• [205423, 2054221513; 7].• [205603, 2052739603; 7].• [205837, 2052976837; 7].

• [206237, 2062674137; 7].• [207121, 2071292281; 7].• [207409, 7089207409; 7].• [207869, 2078414969; 7].• [208283, 2087398283; 7].• [208283, 2087398283; 7].• [208409, 2084768909; 7].• [208657, 4614208657; 7].• [209381, 2099915381; 7].• [210193, 2523810193; 7].

54


• [210317, 2103504017; 7].• [211049, 2110454609; 7].• [211247, 2903911247; 7].• [211873, 2837611873; 7].• [211997, 2119912727; 7].• [212671, 2126714347; 7].• [214007, 2126644007; 7].• [214531, 2147524531; 7].• [214531, 2147524531; 7].• [215617, 2178725617; 7].

• [215723, 2159933723; 7].• [216401, 2124036401; 7].• [216641, 2166456881; 7].• [216967, 2162688967; 7].• [217519, 4023217519; 7].• [217577, 2175507077; 7].• [217793, 2177987543; 7].• [219523, 9624219523; 7].• [219689, 2196871379; 7].• [219721, 2197922821; 7].

• [220663, 2255200663; 7].• [221021, 2881121021; 7].• [221509, 2215522009; 7].• [222043, 2221827043; 7].• [222511, 2870022511; 7].• [223273, 2553823273; 7].• [223331, 2233965731; 7].• [223921, 2239216891; 7].• [223921, 2239216891; 7].• [224233, 2242660933; 7].

• [227459, 4131227459; 7].• [227593, 2159927593; 7].• [228443, 6801228443; 7].• [228793, 2287938613; 7].• [228793, 2287938613; 7].• [229139, 2459329139; 7].• [229321, 5967229321; 7].• [229703, 2297775503; 7].• [229849, 2553229849; 7].• [229849, 2553229849; 7].

• [230273, 2351030273; 7].• [230273, 2351030273; 7].• [230873, 2746130873; 7].• [231559, 2390691559; 7].• [232459, 2324599339; 7].• [232459, 2324599339; 7].• [233327, 2465333327; 7].• [233617, 2336183167; 7].• [234239, 1797234239; 7].• [235181, 2351350181; 7].

• [235181, 2351350181; 7].• [237361, 2373618259; 7].• [237707, 2377097717; 7].• [238373, 2381449373; 7].• [238481, 4260238481; 7].• [238747, 8130238747; 7].• [239171, 8952239171; 7].• [239441, 8031239441; 7].• [239489, 2591639489; 7].• [239753, 2397539561; 7].

• [239831, 2395121831; 7].• [240287, 2318940287; 7].• [240437, 2404333787; 7].• [240587, 2883840587; 7].• [241261, 2772841261; 7].• [241873, 2810941873; 7].• [241951, 2419325851; 7].• [242807, 2741042807; 7].• [243121, 2464533121; 7].• [244787, 2447303687; 7].

• [245171, 2451866471; 7].• [247193, 2115547193; 7].• [247451, 2478622451; 7].• [248293, 3942248293; 7].• [248293, 2482917043; 7].• [248609, 5520248609; 7].• [249797, 2411229797; 7].• [250013, 2500143593; 7].• [250037, 2574100037; 7].• [250433, 2504331539; 7].

• [252709, 2527014019; 7].• [252823, 2528242393; 7].• [253679, 2533465679; 7].• [253819, 2538191017; 7].• [254447, 1734254447; 7].• [254887, 2548637587; 7].• [255149, 2554959149; 7].• [255709, 2551401709; 7].• [256643, 2562544643; 7].• [257489, 2574917489; 7].

• [257489, 2574917489; 7].• [257489, 2574917489; 7].• [257627, 2150057627; 7].• [258067, 2580678943; 7].• [258109, 2581614709; 7].• [258487, 2536008487; 7].• [258787, 2587433587; 7].• [259603, 2200459603; 7].• [260363, 2603623433; 7].• [260809, 2644160809; 7].

• [260809, 2644160809; 7].• [260863, 2608663003; 7].• [261071, 2793861071; 7].• [261407, 2614650107; 7].• [261557, 8997261557; 7].• [261631, 2641461631; 7].• [261631, 2641461631; 7].• [263209, 2632031959; 7].• [263401, 2631878401; 7].• [263759, 2637525269; 7].

• [263827, 2638906027; 7].• [263909, 2639297009; 7].• [264179, 2646105179; 7].• [265381, 2655841381; 7].• [267959, 2679592067; 7].• [268153, 2689405153; 7].• [269473, 2694228373; 7].• [269579, 2695276379; 7].• [270653, 2797890653; 7].• [271849, 2895271849; 7].

• [271849, 2895271849; 7].• [271903, 2776861903; 7].• [272231, 2722313747; 7].• [272927, 2723462927; 7].• [272927, 2723462927; 7].• [273193, 2731938247; 7].• [274457, 1899274457; 7].• [274679, 2746759979; 7].• [274679, 2746759979; 7].• [274739, 2770084739; 7].

• [275227, 2892275227; 7].• [275227, 2892275227; 7].• [275447, 7827275447; 7].• [276079, 2760719419; 7].• [276091, 2761341091; 7].• [276461, 3171276461; 7].• [278329, 2766518329; 7].• [278393, 2254478393; 7].

• [278563, 2766008563; 7].• [278813, 2788171073; 7].• [280069, 2894920069; 7].• [281581, 2879201581; 7].• [281833, 2812805833; 7].• [282011, 2820112523; 7].• [282881, 2826618881; 7].• [283079, 2873683079; 7].• [283079, 2873683079; 7].• [283511, 2833303511; 7].

• [283511, 2833303511; 7].• [284129, 2841978929; 7].• [284743, 2881724743; 7].• [284759, 2837954759; 7].• [285643, 9675285643; 7].• [285841, 4560285841; 7].• [286019, 2897176019; 7].• [286703, 2867058413; 7].• [287347, 2873468497; 7].• [287383, 2872718383; 7].

55


• [287783, 2230487783; 7].• [287801, 2878455401; 7].• [288851, 2887584851; 7].• [289841, 2733589841; 7].• [290659, 2906980159; 7].• [290669, 2929670669; 7].• [291013, 2910223213; 7].• [291257, 2912526857; 7].• [291257, 2912526857; 7].• [291901, 4026291901; 7].

• [292441, 2924493991; 7].• [293099, 2932715099; 7].• [293357, 2826293357; 7].• [293357, 2826293357; 7].• [293863, 2929133863; 7].• [294499, 2749194499; 7].• [294659, 2912184659; 7].• [295459, 2954629759; 7].• [296489, 2975816489; 7].• [296801, 2968394501; 7].

• [296983, 1149296983; 7].• [297421, 2616897421; 7].• [298013, 2282698013; 7].• [298477, 2952928477; 7].• [298651, 2986594651; 7].• [298651, 2986594651; 7].• [298651, 2986594651; 7].• [299447, 2283899447; 7].• [299569, 2816099569; 7].• [300229, 3006054229; 7].

• [300277, 3002779471; 7].• [300493, 3062610493; 7].• [301901, 3534001901; 7].• [303917, 3038790917; 7].• [304013, 3047500013; 7].• [304417, 3065104417; 7].• [304429, 3044295181; 7].• [305533, 3055857733; 7].• [306419, 3176406419; 7].• [306689, 3061002689; 7].

• [307121, 3071294921; 7].• [307121, 3071294921; 7].• [307939, 3079321309; 7].• [309671, 3096716027; 7].• [310439, 3104398841; 7].• [311869, 3161261869; 7].• [312023, 3120238859; 7].• [312203, 3257112203; 7].• [313133, 3131321393; 7].• [313969, 3139851469; 7].

• [314827, 3897014827; 7].• [315803, 3158037473; 7].• [315803, 3158037473; 7].• [316201, 3162014209; 7].• [317411, 3517517411; 7].• [317887, 3178879063; 7].• [318691, 3133308691; 7].• [318701, 3187014953; 7].• [319477, 3192739477; 7].• [319477, 3192739477; 7].• [319547, 3136129547; 7].• [320269, 1776320269; 7].

• [320293, 3202465393; 7].• [321073, 3210228673; 7].• [321091, 3214518091; 7].• [321617, 2394321617; 7].• [321617, 3216174947; 7].• [321617, 3216174947; 7].• [322649, 3226474679; 7].• [322747, 3227448457; 7].• [323249, 3231416249; 7].• [323419, 3234345619; 7].

• [324211, 3246882211; 7].• [324697, 3226074697; 7].• [325709, 3215125709; 7].• [325709, 3215125709; 7].• [328277, 3282177677; 7].• [329597, 1089329597; 7].• [330199, 3306795199; 7].• [330653, 3306968153; 7].• [331171, 3311744881; 7].• [331339, 3311191339; 7].

• [331339, 3311191339; 7].• [331651, 3932431651; 7].• [332699, 3326751599; 7].• [333041, 2832333041; 7].• [333269, 3332643329; 7].• [334231, 3342395191; 7].• [334877, 5328334877; 7].• [334993, 9327334993; 7].• [336307, 3325896307; 7].• [336527, 3316506527; 7].

• [337153, 3814237153; 7].• [337361, 4776337361; 7].• [338449, 6216338449; 7].• [338449, 3385994449; 7].• [339827, 3333279827; 7].• [339827, 3398262047; 7].• [340057, 3400587667; 7].• [340117, 6204340117; 7].• [342061, 3420690151; 7].• [342607, 3426065557; 7].

• [343687, 3436881067; 7].• [343771, 8874343771; 7].• [344347, 3466514347; 7].• [344759, 3447522629; 7].• [344857, 1956344857; 7].• [345089, 3927645089; 7].• [345307, 3786945307; 7].• [349109, 3491036909; 7].• [349331, 3493398191; 7].• [349813, 6096349813; 7].

• [351343, 3515871343; 7].• [351343, 3515871343; 7].• [353099, 9831353099; 7].• [353681, 3536816489; 7].• [354539, 3547332539; 7].• [354661, 3541089661; 7].• [354743, 4995354743; 7].• [355009, 3550429009; 7].• [355483, 1452355483; 7].• [355573, 3552118573; 7].

• [356647, 3579206647; 7].• [357359, 3216657359; 7].• [357977, 3571023977; 7].• [358219, 3582527119; 7].• [358769, 3587610239; 7].• [359327, 3748859327; 7].• [359987, 3518549987; 7].• [360271, 3602263471; 7].• [360637, 3605874637; 7].• [360653, 3654330653; 7].

• [361217, 3612462017; 7].• [361663, 3232861663; 7].• [362309, 9528362309; 7].• [364267, 3642914167; 7].• [365107, 6162365107; 7].• [365179, 3675245179; 7].• [365749, 3651596749; 7].• [366097, 8349366097; 7].• [366859, 3668804059; 7].• [367019, 3670194929; 7].

• [367019, 3670194929; 7].• [368647, 3929168647; 7].• [368911, 3659528911; 7].• [369013, 3141969013; 7].• [369407, 3609069407; 7].• [369557, 3695634857; 7].• [370673, 3786160673; 7].• [371353, 3809471353; 7].• [372013, 3720846313; 7].• [372653, 3743232653; 7].

56


• [372833, 3198672833; 7].• [372877, 3728932777; 7].• [373937, 3739898537; 7].• [373969, 3739205269; 7].• [374149, 3748343149; 7].• [374987, 6063374987; 7].• [375097, 3750777097; 7].• [375127, 3725505127; 7].• [375367, 3753697237; 7].• [375631, 3790125631; 7].

• [375703, 8469375703; 7].• [376021, 3760298371; 7].• [376049, 3760188149; 7].• [376351, 7785376351; 7].• [377477, 3774483077; 7].• [377513, 4251377513; 7].• [377563, 6567377563; 7].• [378439, 3733068439; 7].• [378691, 3784095691; 7].• [379009, 3790076779; 7].

• [379877, 3799783877; 7].• [380713, 3366080713; 7].• [381419, 3816012419; 7].• [382583, 3928882583; 7].• [384407, 3844834907; 7].• [385393, 3723685393; 7].• [386039, 3868538039; 7].• [386117, 3814766117; 7].• [388211, 3275488211; 7].• [388259, 3887977259; 7].

• [389437, 1971389437; 7].• [389533, 3895361713; 7].• [389749, 3594389749; 7].• [389749, 3594389749; 7].• [390263, 6879390263; 7].• [390893, 3908470193; 7].• [391231, 3912348841; 7].• [391351, 3918028351; 7].• [391757, 3632491757; 7].• [392737, 3927587437; 7].

• [392927, 3913922927; 7].• [392963, 3940202963; 7].• [394861, 3948635311; 7].• [395537, 5928395537; 7].• [395537, 3955844537; 7].• [395537, 3955844537; 7].• [396041, 3960306041; 7].• [396041, 3960306041; 7].• [398393, 3935378393; 7].• [399379, 3993755989; 7].

• [399409, 3994090489; 7].• [399577, 3995916577; 7].• [399577, 3995916577; 7].• [399989, 3999917489; 7].• [400307, 4076590307; 7].• [400759, 4007592943; 7].• [401017, 4270001017; 7].• [401711, 5106401711; 7].• [402551, 4025295551; 7].• [402551, 4025295551; 7].

• [402797, 4027255997; 7].• [402869, 4023897869; 7].• [403163, 4071673163; 7].• [403363, 4039357363; 7].• [403553, 4031182553; 7].• [407263, 4077761263; 7].• [408209, 4089054209; 7].• [408997, 4070688997; 7].• [410119, 4101198187; 7].• [410717, 4167830717; 7].

• [411361, 4143711361; 7].• [411361, 4143711361; 7].• [412303, 4440712303; 7].• [412891, 4128935281; 7].• [413089, 1410413089; 7].• [413711, 4137683111; 7].• [414097, 8982414097; 7].• [414457, 4144259557; 7].• [416419, 4164604219; 7].• [417133, 2283417133; 7].

• [417577, 4175886277; 7].• [417581, 4450017581; 7].• [418783, 4187838493; 7].• [418783, 4187838493; 7].• [418867, 4144648867; 7].• [419401, 4193384401; 7].• [419921, 4963019921; 7].• [420457, 4876620457; 7].• [420557, 4205574299; 7].• [420779, 4203927779; 7].

• [422459, 4224591827; 7].• [423097, 4230920557; 7].• [423439, 4234338859; 7].• [423823, 4784223823; 7].• [423883, 4238911783; 7].• [423931, 4239376831; 7].• [423931, 4239376831; 7].• [425317, 4253195707; 7].• [426061, 4261860061; 7].• [426383, 4263832613; 7].

• [426383, 4263832613; 7].• [426799, 4220076799; 7].• [427079, 4270986179; 7].• [427919, 4842727919; 7].• [429161, 4552629161; 7].• [430699, 4343980699; 7].• [430753, 4306225753; 7].• [431441, 4314492761; 7].• [431657, 4719131657; 7].• [432037, 4320417337; 7].

• [432343, 9720432343; 7].• [432893, 4328226593; 7].• [433093, 4330938427; 7].• [433889, 4338850679; 7].• [433967, 4339688417; 7].• [434179, 4341780259; 7].• [434867, 4444034867; 7].• [434939, 4349352869; 7].• [435151, 4351601251; 7].• [435191, 4351319591; 7].

• [435481, 4354897951; 7].• [435541, 4912335541; 7].• [436621, 4361476621; 7].• [436621, 4361476621; 7].• [439723, 4397286463; 7].• [440653, 4406538757; 7].• [440753, 4407816653; 7].• [442577, 2652442577; 7].• [443419, 4334543419; 7].• [443753, 4437540143; 7].

• [445771, 4455667771; 7].• [446197, 4464935197; 7].• [446839, 9228446839; 7].• [447179, 4474653179; 7].• [449083, 5667449083; 7].• [449243, 4498364243; 7].• [449653, 4403549653; 7].• [449951, 4499517773; 7].• [450383, 4503596183; 7].• [451013, 4389451013; 7].

• [451013, 4389451013; 7].• [451279, 4512768319; 7].• [451699, 4516969729; 7].• [451901, 4519060841; 7].• [452009, 4520093579; 7].• [452009, 4520093579; 7].• [453659, 4839153659; 7].• [453889, 4538806189; 7].• [454721, 5712454721; 7].• [455353, 4553551273; 7].

• [455443, 4980155443; 7].• [455711, 4282055711; 7].• [455717, 4557175433; 7].• [457189, 4571895067; 7].• [457547, 4575413417; 7].• [458729, 4583642729; 7].• [458963, 4565228963; 7].• [459127, 5703459127; 7].• [460451, 8778460451; 7].• [461521, 4283861521; 7].

57


• [462263, 4555662263; 7].• [462263, 4684662263; 7].• [462263, 4684662263; 7].• [466201, 4539166201; 7].• [466261, 4662625801; 7].• [469351, 9339469351; 7].• [469649, 4697893649; 7].• [470521, 4702598521; 7].• [470653, 4706515423; 7].• [472273, 4722767383; 7].

• [472369, 4723687789; 7].• [473027, 4795043027; 7].• [473101, 4955873101; 7].• [474043, 9162474043; 7].• [474413, 4744134779; 7].• [475273, 7728475273; 7].• [476591, 4765353191; 7].• [476753, 3960476753; 7].• [476863, 4769342863; 7].• [477517, 1470477517; 7].

• [478451, 4784528321; 7].• [478801, 1512478801; 7].• [481177, 4811774269; 7].• [481181, 4811800181; 7].• [481181, 4811800181; 7].• [481697, 4818661697; 7].• [481697, 4818661697; 7].• [482641, 4926482641; 7].• [482641, 4926482641; 7].• [482899, 4828942399; 7].

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• [490031, 4902268031; 7].• [490537, 4905372181; 7].• [491003, 4976771003; 7].• [491461, 4957271461; 7].• [491501, 4911584501; 7].• [493013, 4881193013; 7].• [493219, 4984153219; 7].• [495749, 4968045749; 7].• [495967, 4250895967; 7].• [495967, 4343895967; 7].

• [496687, 4966875493; 7].• [497069, 4970899169; 7].• [497339, 4941917339; 7].• [498397, 4983984157; 7].• [499633, 4936429633; 7].• [499729, 4997273299; 7].• [501019, 5016339019; 7].• [501191, 6786501191; 7].• [502261, 5233102261; 7].• [502781, 5021353781; 7].

• [502861, 2547502861; 7].• [502973, 9261502973; 7].• [503303, 5033037599; 7].• [505061, 5552305061; 7].• [506269, 5062679059; 7].• [506461, 5041766461; 7].• [506647, 5357606647; 7].• [506837, 5068155737; 7].• [507673, 5076493273; 7].• [507719, 5077193477; 7].

• [507839, 5079111839; 7].• [509513, 5036249513; 7].• [510179, 5539110179; 7].• [510319, 5103185929; 7].• [510379, 5103771919; 7].• [511013, 7293511013; 7].• [511417, 5120221417; 7].• [511463, 5114315363; 7].• [512717, 5127123377; 7].• [514117, 5165644117; 7].

• [514247, 5143666247; 7].• [514357, 5141404357; 7].• [514357, 5141404357; 7].• [514433, 8490514433; 7].• [514453, 5517214453; 7].• [514757, 5147563697; 7].• [515887, 5158659487; 7].• [516449, 5128266449; 7].• [516973, 5167788973; 7].• [517999, 4167517999; 7].

• [518191, 5181952381; 7].• [518291, 5182916369; 7].• [518299, 5111018299; 7].• [518299, 5111018299; 7].• [518299, 5182938829; 7].• [519881, 5134539881; 7].• [520297, 5202196297; 7].• [520297, 5202196297; 7].• [522017, 5297382017; 7].• [522959, 5853222959; 7].

• [524893, 2577524893; 7].• [526717, 5502526717; 7].• [526717, 5502526717; 7].• [526739, 5267433539; 7].• [527897, 5552327897; 7].• [528013, 5280183523; 7].• [528317, 5283176753; 7].• [528373, 5227368373; 7].• [529033, 5521429033; 7].• [530983, 5309839081; 7].

• [531877, 5318806177; 7].• [532349, 5323239149; 7].• [532501, 5130832501; 7].• [532849, 8358532849; 7].• [533809, 5338712809; 7].• [533809, 5338712809; 7].• [534581, 5345819351; 7].• [534581, 5345819351; 7].• [535193, 5351299193; 7].• [535193, 5351299193; 7].

• [535481, 5386135481; 7].• [535481, 5386135481; 7].• [535727, 5357501027; 7].• [536057, 7326536057; 7].• [536849, 3066536849; 7].• [537079, 5255937079; 7].• [537157, 5373531157; 7].• [537847, 3405537847; 7].• [538841, 5383294841; 7].• [539899, 5398961899; 7].

• [539899, 5398961899; 7].• [539899, 5398961899; 7].• [540509, 8355540509; 7].• [540691, 4254540691; 7].• [541859, 5418387959; 7].• [543143, 5437089143; 7].• [543289, 5370243289; 7].• [543307, 5434767307; 7].• [544399, 5443747999; 7].• [544759, 5477584759; 7].

• [545387, 5750645387; 7].• [545473, 5378045473; 7].• [545533, 5455336819; 7].• [545647, 5456465557; 7].• [546661, 5361546661; 7].• [546661, 5361546661; 7].• [547369, 5473657339; 7].• [548003, 4056548003; 7].• [548657, 2370548657; 7].• [548827, 5426648827; 7].

58


• [548827, 5426648827; 7].• [549203, 5412399203; 7].• [549431, 5494766531; 7].• [549511, 3729549511; 7].• [549569, 5327349569; 7].• [549883, 9252549883; 7].• [549937, 5499224737; 7].• [550861, 5508822661; 7].• [550997, 5509974653; 7].• [552001, 7056552001; 7].

• [553733, 5597833733; 7].• [555041, 5558010041; 7].• [555349, 5553482719; 7].• [555593, 5555913533; 7].• [555827, 2496555827; 7].• [557021, 5860157021; 7].• [558203, 5558748203; 7].• [559577, 5596681577; 7].• [559973, 5404159973; 7].• [560237, 5202860237; 7].

• [560797, 5895260797; 7].• [560887, 5608336687; 7].• [561559, 5615599597; 7].• [561961, 5619670171; 7].• [562421, 5648512421; 7].• [562621, 5621980621; 7].• [563051, 1302563051; 7].• [565039, 5650399087; 7].• [566617, 5670956617; 7].• [566693, 5666960063; 7].

• [567011, 5624367011; 7].• [567011, 5624367011; 7].• [568033, 5680336699; 7].• [568783, 5654488783; 7].• [570961, 3933570961; 7].• [571019, 5492371019; 7].• [571147, 5711536147; 7].• [571147, 5711536147; 7].• [571229, 5716447229; 7].• [571369, 5593471369; 7].

• [572479, 5724786619; 7].• [572573, 7869572573; 7].• [574297, 9177574297; 7].• [574933, 9624574933; 7].• [576431, 5764891031; 7].• [578251, 5782812151; 7].• [578777, 9933578777; 7].• [579521, 5795214041; 7].• [579521, 5795214041; 7].• [579773, 5731419773; 7].• [581429, 7854581429; 7].• [582227, 5821134227; 7].

• [582319, 5823167269; 7].• [583603, 5836435603; 7].• [583603, 5836435603; 7].• [584561, 5845618607; 7].• [585601, 7539585601; 7].• [586361, 5863612907; 7].• [587549, 3186587549; 7].• [588097, 5895288097; 7].• [588097, 5895288097; 7].• [589607, 3273589607; 7].

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• [603349, 6033496621; 7].• [603443, 6070713443; 7].• [603541, 6039423541; 7].• [603541, 6039423541; 7].• [603553, 6054093553; 7].• [603917, 6212103917; 7].• [604379, 9198604379; 7].• [605009, 6050035559; 7].• [605039, 6050373419; 7].• [606569, 6065446169; 7].

• [606643, 6066800143; 7].• [606653, 1002606653; 7].• [607163, 6966007163; 7].• [608371, 6081221371; 7].• [609149, 6094992149; 7].• [610193, 6105310193; 7].• [610193, 6105310193; 7].• [610339, 1320610339; 7].• [610781, 9240610781; 7].• [611561, 2076611561; 7].

• [612037, 6990012037; 7].• [614143, 6402914143; 7].• [614293, 6819614293; 7].• [614293, 6819614293; 7].• [614659, 6146544409; 7].• [614881, 6148851361; 7].• [615473, 6929415473; 7].• [616181, 6306016181; 7].• [616289, 6162645389; 7].• [617657, 9756617657; 7].

• [619139, 6198895139; 7].• [619187, 1539619187; 7].• [620467, 6389420467; 7].• [620603, 6206092973; 7].• [621059, 6354621059; 7].• [621059, 6354621059; 7].• [621083, 6219831083; 7].• [621083, 6219831083; 7].• [621697, 6346821697; 7].• [622397, 6223987097; 7].

• [622397, 6223987097; 7].• [622399, 2772622399; 7].• [623531, 6248363531; 7].• [623669, 6236640629; 7].• [623957, 6889523957; 7].• [623989, 6277253989; 7].• [624509, 6245016599; 7].• [624787, 6927024787; 7].• [624983, 6241089983; 7].• [625007, 6250098737; 7].

• [628781, 6284323781; 7].• [628909, 6290158909; 7].• [629011, 6290807311; 7].• [630919, 6309941719; 7].• [631643, 6318247643; 7].• [632081, 6320432381; 7].• [632351, 6323593571; 7].• [632993, 6323993993; 7].• [633823, 2619633823; 7].• [634141, 6341956741; 7].

• [634169, 6341320169; 7].• [634169, 6341320169; 7].• [634421, 6342046421; 7].• [634649, 6166834649; 7].• [636073, 8019636073; 7].• [641279, 6498821279; 7].• [641581, 6415277881; 7].• [642881, 6428814683; 7].• [642953, 8316642953; 7].• [643073, 1167643073; 7].

59


• [643589, 6435853979; 7].• [643729, 6528643729; 7].• [643729, 6528643729; 7].• [644629, 6446271739; 7].• [644869, 4188644869; 7].• [647189, 6471896429; 7].• [647189, 6471896429; 7].• [647741, 6477343541; 7].• [648649, 6487848649; 7].• [648649, 6487848649; 7].

• [648649, 6487848649; 7].• [648859, 6491988859; 7].• [649261, 6472099261; 7].• [649471, 2706649471; 7].• [649631, 6496393451; 7].• [650011, 3168650011; 7].• [652279, 6522293779; 7].• [653057, 6578423057; 7].• [653503, 6534113503; 7].• [653503, 6534113503; 7].

• [654343, 6541782343; 7].• [654421, 6544232791; 7].• [655003, 6558367003; 7].• [655601, 6556713101; 7].• [656141, 6561362141; 7].• [656141, 6561362141; 7].• [657703, 6577057363; 7].• [658831, 6588384571; 7].• [659189, 6592928189; 7].• [660601, 6606070201; 7].

• [660809, 6304560809; 7].• [663547, 6635478853; 7].• [664271, 6642724901; 7].• [664549, 6591664549; 7].• [664549, 6591664549; 7].• [665251, 6654476251; 7].• [665857, 6659972857; 7].• [666143, 6661432601; 7].• [666467, 6744066467; 7].• [669377, 6693179177; 7].

• [670447, 6701560447; 7].• [670627, 6709300627; 7].• [673019, 6730849919; 7].• [673613, 6736176383; 7].• [673781, 6737813633; 7].• [674731, 6747409231; 7].• [675067, 6750689287; 7].• [675347, 6744465347; 7].• [676903, 6769176703; 7].• [677371, 6402077371; 7].

• [679319, 6793215719; 7].• [680027, 6837680027; 7].• [680027, 6837680027; 7].• [680027, 6837680027; 7].• [680569, 6845720569; 7].• [681449, 6814696349; 7].• [682069, 6820690399; 7].• [683461, 6834175861; 7].• [683681, 6836842151; 7].• [683911, 4551683911; 7].

• [684053, 6827604053; 7].• [684637, 6846353827; 7].• [685393, 6855760393; 7].• [686321, 6810116321; 7].• [686801, 6868177601; 7].• [687461, 6874615607; 7].• [688249, 6882466639; 7].• [688939, 6886084939; 7].• [689089, 6890726989; 7].• [689693, 6896958413; 7].

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• [706313, 7034686313; 7].• [706943, 7293706943; 7].• [706943, 7293706943; 7].• [707117, 4257707117; 7].• [707177, 7071811277; 7].• [707467, 7074854467; 7].• [707467, 7074854467; 7].• [707527, 7075876627; 7].• [707543, 6087707543; 7].• [708473, 7084787933; 7].

• [708569, 5556708569; 7].• [709231, 7976809231; 7].• [709843, 7089679843; 7].• [710623, 7105544623; 7].• [712693, 7165432693; 7].• [712847, 7128493217; 7].• [712961, 7128889961; 7].• [713399, 2271713399; 7].• [713407, 7297313407; 7].• [714677, 7146727517; 7].

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• [731623, 7575131623; 7].• [732209, 7805232209; 7].• [732467, 5763732467; 7].• [733099, 7333498099; 7].• [733883, 7338595883; 7].• [733883, 7338595883; 7].• [734647, 7346457757; 7].• [734879, 7348463279; 7].• [735143, 7351557743; 7].• [735617, 7356141887; 7].

• [735901, 7359023701; 7].• [736037, 7360349417; 7].• [736063, 7360626463; 7].• [736063, 7360626463; 7].• [736441, 7330276441; 7].• [737927, 7377416927; 7].• [740671, 7407557671; 7].• [741347, 7413645947; 7].• [743881, 7125443881; 7].• [744371, 7445544371; 7].

• [744371, 7445544371; 7].• [744371, 7445544371; 7].• [744599, 7445995313; 7].• [746107, 7461028717; 7].• [747521, 3942747521; 7].• [749069, 8526749069; 7].• [750487, 7293750487; 7].• [750487, 7293750487; 7].• [750857, 7508578001; 7].• [751759, 7517978359; 7].

60


• [751879, 7518763939; 7].• [752117, 7521912017; 7].• [752821, 7528230001; 7].• [757993, 1911757993; 7].• [758363, 8124758363; 7].• [759113, 7591132229; 7].• [760579, 7624660579; 7].• [760579, 7624660579; 7].• [760913, 7609169993; 7].• [761009, 7612277009; 7].

• [762031, 3021762031; 7].• [762791, 7627151891; 7].• [763999, 7639994827; 7].• [764293, 7676294293; 7].• [766387, 7664635387; 7].• [767881, 7678204081; 7].• [769661, 7696715261; 7].• [769919, 8652769919; 7].• [771481, 7713588481; 7].• [771643, 7716455953; 7].

• [771997, 7840971997; 7].• [772657, 7719982657; 7].• [773117, 7225373117; 7].• [773501, 7735060601; 7].• [773821, 5241773821; 7].• [774779, 7591074779; 7].• [775573, 7945075573; 7].• [775669, 6657775669; 7].• [778357, 6864778357; 7].• [779173, 3066779173; 7].

• [779663, 7993679663; 7].• [780179, 9033780179; 7].• [780953, 7719780953; 7].• [780953, 7719780953; 7].• [781051, 7857811051; 7].• [781271, 2805781271; 7].• [782267, 7824575267; 7].• [783257, 7832521217; 7].• [783599, 7832838599; 7].• [784423, 7847854423; 7].

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• [803717, 8570303717; 7].• [804919, 8049140179; 7].• [804997, 8549304997; 7].• [805241, 8604205241; 7].• [806041, 8066402041; 7].• [807487, 8074836967; 7].• [808039, 3081808039; 7].• [810379, 8413710379; 7].• [810419, 8102169419; 7].• [811147, 8111459017; 7].

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• [847129, 8446417129; 7].• [849347, 8493471893; 7].• [849833, 8498339543; 7].• [849833, 8498339543; 7].• [851507, 8868251507; 7].• [851723, 8517297083; 7].• [851839, 7161851839; 7].• [852557, 5259852557; 7].• [853439, 8533351439; 7].• [854419, 8576384419; 7].

• [854683, 8546499883; 7].• [855067, 8550702367; 7].• [857099, 8599457099; 7].• [857099, 8599457099; 7].• [857137, 8575196137; 7].• [857249, 8542247249; 7].• [857267, 5229857267; 7].• [860939, 8605841939; 7].• [861037, 8610348457; 7].• [863003, 8678483003; 7].

• [863729, 8669003729; 7].• [864509, 8849064509; 7].• [866623, 8666051623; 7].• [867743, 8679306743; 7].• [869371, 8693744011; 7].• [869443, 8694509443; 7].• [869443, 8694509443; 7].• [869443, 8694509443; 7].• [870593, 8728590593; 7].• [871289, 8712833009; 7].

61


• [871883, 4620871883; 7].• [872471, 8674172471; 7].• [872563, 8723405563; 7].• [874037, 8740320167; 7].• [874771, 8747990971; 7].• [874987, 8523574987; 7].• [875519, 8755159319; 7].• [875519, 8755159319; 7].• [877301, 9027877301; 7].• [877301, 8773041581; 7].

• [877313, 8773157213; 7].• [877313, 8773157213; 7].• [881173, 8811764503; 7].• [881963, 8811311963; 7].• [881963, 8811311963; 7].• [882169, 8821667689; 7].• [883343, 8833933343; 7].• [883343, 8833933343; 7].• [883343, 8833933343; 7].• [884717, 8847285917; 7].

• [885163, 8851786663; 7].• [885503, 8855088653; 7].• [885589, 8855135689; 7].• [885893, 8857863893; 7].• [885991, 8853063991; 7].• [886199, 8861508899; 7].• [886381, 8714986381; 7].• [888541, 8770788541; 7].• [888637, 8813508637; 7].• [888967, 8468688967; 7].

• [889901, 8180389901; 7].• [891151, 8812191151; 7].• [892249, 8935972249; 7].• [892709, 8948182709; 7].• [893407, 3756893407; 7].• [893657, 8187893657; 7].• [893657, 8187893657; 7].• [893681, 8936164781; 7].• [894227, 8942933327; 7].• [894689, 8946805289; 7].

• [895051, 8977975051; 7].• [896327, 8964479327; 7].• [896521, 8965285921; 7].• [896521, 8965285921; 7].• [897163, 8544597163; 7].• [899891, 8998945841; 7].• [900409, 9073080409; 7].• [900551, 9005533631; 7].• [901547, 9014479547; 7].• [902311, 9023283811; 7].

• [902723, 9025355723; 7].• [904811, 1851904811; 7].• [905551, 9051014551; 7].• [906107, 9337506107; 7].• [906391, 9944706391; 7].• [906767, 9066654767; 7].• [906847, 9068473663; 7].• [907163, 9071065163; 7].• [908533, 9098528533; 7].• [908653, 9012878653; 7].

• [909791, 9097351691; 7].• [910781, 9107393681; 7].• [911413, 2400911413; 7].• [914723, 5748914723; 7].• [914857, 9148621357; 7].• [915727, 9759015727; 7].• [917591, 9175995371; 7].• [917617, 9176153317; 7].• [917617, 9176153317; 7].• [918011, 9180672311; 7].

• [919013, 9331519013; 7].• [920741, 5526920741; 7].• [920971, 9256610971; 7].• [923233, 9231122233; 7].• [923513, 9643823513; 7].• [924083, 9823224083; 7].• [925597, 9287425597; 7].• [925597, 9287425597; 7].• [926129, 6525926129; 7].• [926411, 9262925411; 7].

• [927517, 4521927517; 7].• [928177, 9281881177; 7].• [928177, 9281881177; 7].• [928819, 7029928819; 7].• [929129, 9291267989; 7].• [930571, 9354600571; 7].• [931193, 9314257193; 7].• [932303, 9323531303; 7].• [932303, 9323531303; 7].• [932593, 9325155193; 7].

• [932681, 9323144681; 7].• [932947, 9323366947; 7].• [935699, 9280535699; 7].• [936181, 9213036181; 7].• [936451, 8211936451; 7].• [936737, 9069936737; 7].• [936967, 9369697147; 7].• [937243, 7716937243; 7].• [937571, 9377593571; 7].• [938533, 9389135533; 7].

• [939611, 9394356611; 7].• [940483, 9455080483; 7].• [940691, 9406895591; 7].• [941981, 9474911981; 7].• [942101, 3534942101; 7].• [943637, 9436318247; 7].• [944071, 9441401071; 7].• [945937, 9459348727; 7].• [945961, 3327945961; 7].• [946037, 2961946037; 7].

• [946769, 9467657249; 7].• [947411, 9478799411; 7].• [947501, 9489707501; 7].• [948593, 9483219593; 7].• [949159, 9394849159; 7].• [949889, 9494251889; 7].• [949997, 9499756997; 7].• [949997, 9499756997; 7].• [950029, 9687950029; 7].• [950029, 9687950029; 7].

• [950083, 9500840863; 7].• [950111, 9565490111; 7].• [950953, 9591830953; 7].• [952547, 9525903347; 7].• [953497, 9549173497; 7].• [953567, 9581063567; 7].• [954971, 9811854971; 7].• [955819, 9558172879; 7].• [957181, 9571410181; 7].• [957181, 9571410181; 7].

• [959363, 9729359363; 7].• [960521, 7629960521; 7].• [960677, 9606741887; 7].• [962783, 9627135383; 7].• [963601, 9636010231; 7].• [964507, 9645023617; 7].• [965491, 9656507491; 7].• [966233, 9662363033; 7].• [966233, 9662363033; 7].• [966373, 9663225973; 7].

• [967229, 9672310229; 7].• [967229, 9672310229; 7].• [967333, 9673796533; 7].• [967511, 9677968511; 7].• [968519, 9762368519; 7].• [970027, 6015970027; 7].• [970213, 9743200213; 7].• [970877, 9718330877; 7].• [970927, 9709282117; 7].• [970967, 9709663787; 7].

• [972373, 9726741373; 7].• [973187, 9719653187; 7].• [974159, 9479774159; 7].• [974489, 3165974489; 7].• [974861, 9747434861; 7].• [974861, 9747434861; 7].• [975053, 9750521153; 7].• [975053, 9750521153; 7].• [975629, 9756151829; 7].

62


• [976489, 9764838289; 7].• [976489, 9764838289; 7].• [977023, 9713327023; 7].• [977357, 9773205257; 7].• [977719, 9777336619; 7].• [978079, 9786897079; 7].• [979361, 9793690481; 7].• [979709, 9792664709; 7].• [979757, 2046979757; 7].

• [980327, 9803272991; 7].• [981527, 9815749127; 7].• [982021, 9820258321; 7].• [982021, 9820258321; 7].• [982841, 9487282841; 7].• [982981, 9829846261; 7].• [984749, 9847423709; 7].• [984757, 9482784757; 7].• [985997, 1881985997; 7].

• [986113, 3570986113; 7].• [986429, 9877826429; 7].• [987361, 9873617101; 7].• [987361, 9873617101; 7].• [987941, 9879823841; 7].• [988007, 9983488007; 7].• [989173, 9895556173; 7].• [989239, 9892762339; 7].• [990469, 9901443469; 7].

• [991037, 9910379813; 7].• [991387, 9917872387; 7].• [994363, 9943663153; 7].• [994793, 9581194793; 7].• [995381, 9953893811; 7].• [997609, 9958657609; 7].• [998219, 9988370219; 7].• [998743, 9003998743; 7].

5.5 Five Digits RepetitionsBelow are some examples of representations:

• [1 51, 11759551; 7] I 1511 17595 511 17595 17595 511 17595 17595 17595 511 17595 17595 17595 17595 511 17595 17595 17595 17595 17595 511 17595 17595 17595 17595 17595 17595 51.

• [199, 19992283; 7]. I 199199 92283199 92283 92283199 92283 92283 92283199 92283 92283 92283 92283199 92283 92283 92283 92283 92283199 92283 92283 92283 92283 92283 92283.


• [313, 34869013; 7].• [599, 59914229; 7].• [599, 59914229; 7].• [601, 60688081; 7].• [761, 76132061; 7].• [761, 76132061; 7].• [761, 76132061; 7].• [857, 85729409; 7].

• [1013, 101480993; 7].• [1049, 104620679; 7].• [1367, 138063467; 7].• [1709, 436171709; 7].• [1723, 178781923; 7].• [1741, 171737641; 7].• [1783, 178369069; 7].• [1913, 534541913; 7].• [2081, 201347981; 7].• [2207, 235922207; 7].

• [2207, 235922207; 7].• [2273, 227342483; 7].• [2273, 227342483; 7].• [2333, 599552333; 7].• [2371, 238373671; 7].• [2411, 220259411; 7].• [2437, 270800437; 7].• [2473, 247357273; 7].• [2473, 247357273; 7].• [2473, 247357273; 7].

• [2617, 261988057; 7].• [2731, 272891431; 7].• [2833, 286564933; 7].• [2857, 289427857; 7].• [2857, 289427857; 7].• [3041, 383232041; 7].• [3217, 614043217; 7].• [3517, 379494517; 7].• [4079, 407434619; 7].• [4153, 541594153; 7].

63


• [4259, 423555959; 7].• [4493, 446262893; 7].• [4733, 441760733; 7].• [5081, 224975081; 7].• [5443, 577694443; 7].• [5743, 794885743; 7].• [5743, 574771123; 7].• [5813, 720305813; 7].• [5827, 858065827; 7].• [6079, 602817979; 7].

• [6229, 643383229; 7].• [6373, 638028373; 7].• [6373, 638028373; 7].• [6379, 384936379; 7].• [6547, 861096547; 7].• [6679, 138126679; 7].• [7109, 517867109; 7].• [7219, 722402719; 7].• [7243, 724658443; 7].• [7243, 724658443; 7].

• [7333, 578137333; 7].• [7639, 762185839; 7].• [7759, 770989759; 7].• [8011, 845876011; 7].• [8513, 854189813; 7].• [8641, 864190351; 7].• [8641, 864190351; 7].• [8863, 108788863; 7].

• [8969, 896568059; 7].• [9043, 904372741; 7].• [9377, 937357637; 7].• [9439, 948979039; 7].• [9547, 956638447; 7].• [9613, 962109913; 7].• [9859, 298869859; 7].

• [10193, 1877980193; 7].• [10427, 1042922957; 7].• [10529, 1052849309; 7].• [10729, 1072920727; 7].• [10903, 1480900903; 7].• [10973, 1940410973; 7].• [10973, 1940410973; 7].• [11197, 1141394197; 7].• [11489, 1157657489; 7].• [11489, 1148876669; 7].

• [11551, 1312871551; 7].• [11551, 1156153951; 7].• [11689, 1634291689; 7].• [11689, 1168687459; 7].• [11699, 1127780699; 7].• [11821, 1649261821; 7].• [11839, 1120073839; 7].• [12041, 6885012041; 7].• [12301, 1232037901; 7].• [12343, 1237928743; 7].

• [12479, 1490592479; 7].• [13441, 1342738741; 7].• [13711, 5972113711; 7].• [13751, 1375697291; 7].• [13763, 1346827763; 7].• [13883, 6249913883; 7].• [14071, 2941214071; 7].• [14071, 9418514071; 7].• [14243, 1179794243; 7].• [14867, 1489722467; 7].

• [14891, 1416365891; 7].• [15461, 1546926701; 7].• [15541, 1554172387; 7].• [15629, 1562823539; 7].• [15671, 1569482471; 7].• [16061, 1606216571; 7].• [16087, 1608788179; 7].• [16141, 1614443011; 7].• [16229, 1652683229; 7].• [16249, 1658218249; 7].

• [16273, 1627889023; 7].• [16301, 1893706301; 7].• [16333, 1633477303; 7].• [16561, 1656315121; 7].• [16693, 1665946693; 7].• [16693, 1665946693; 7].• [16693, 1665946693; 7].• [16901, 1693810301; 7].• [17737, 1773740653; 7].• [19433, 1943840813; 7].

• [19553, 1947958553; 7].• [19609, 1982404609; 7].• [19979, 1373989979; 7].• [19993, 5536819993; 7].• [20177, 2030834177; 7].• [20183, 2062226183; 7].• [20509, 2054048509; 7].• [20509, 2054048509; 7].• [21019, 2154153019; 7].• [21383, 2138652443; 7].

• [21433, 2981241433; 7].• [21589, 2137611589; 7].• [21589, 2137611589; 7].• [21647, 2166921947; 7].• [21701, 2170138433; 7].• [21727, 2342571727; 7].• [21727, 2172782761; 7].• [21841, 3054621841; 7].• [21961, 2199251161; 7].• [22259, 5217322259; 7].

• [22961, 3897622961; 7].• [23021, 2302129463; 7].• [23041, 2393720041; 7].• [23081, 2306507381; 7].• [23203, 2304773203; 7].• [23333, 2334555233; 7].• [23593, 2359595113; 7].• [23869, 2386712299; 7].• [23887, 2388300547; 7].• [23917, 4645223917; 7].

• [23993, 2322761993; 7].• [24107, 2427729107; 7].• [24337, 2433733117; 7].• [24337, 2433733117; 7].• [24691, 4095324691; 7].• [24989, 2498549969; 7].• [25301, 2570299301; 7].• [25339, 2533757449; 7].• [25799, 2579747099; 7].• [25799, 2579747099; 7].

• [26407, 2640737809; 7].• [26501, 2650413971; 7].• [26627, 2487476627; 7].• [26683, 2785736683; 7].• [26717, 2796386717; 7].• [26953, 2698939153; 7].• [27043, 2752707043; 7].• [27043, 2752707043; 7].• [27067, 2709415867; 7].• [27253, 5424327253; 7].

• [27283, 2723840683; 7].• [28517, 2591428517; 7].• [28517, 2591428517; 7].• [28571, 2843848571; 7].• [28571, 2843848571; 7].• [28663, 2866345879; 7].• [28789, 2878185679; 7].• [28859, 2885917403; 7].• [29017, 2892149017; 7].• [29033, 2927420033; 7].

• [29191, 2978291191; 7].• [29207, 2953940207; 7].• [29231, 9051029231; 7].• [29243, 5527829243; 7].• [29411, 1657529411; 7].• [29803, 2985223903; 7].• [30089, 3009280289; 7].• [30161, 3016520861; 7].• [30161, 3016520861; 7].• [30323, 3032107073; 7].

64


• [30497, 3045517997; 7].• [30727, 3075914827; 7].• [30839, 3073476839; 7].• [31259, 3125554109; 7].• [31547, 3584731547; 7].• [31547, 3584731547; 7].• [31627, 3178363627; 7].• [31817, 6117331817; 7].• [31891, 3194299891; 7].• [32563, 3241613563; 7].

• [32563, 3256738393; 7].• [32573, 3708722573; 7].• [32797, 3276368497; 7].• [33161, 3801963161; 7].• [33289, 3328405819; 7].• [33937, 3532383937; 7].• [34019, 3401242049; 7].• [34583, 8395234583; 7].• [34739, 1371034739; 7].• [35839, 3784295839; 7].

• [36073, 3665424073; 7].• [36083, 3608549333; 7].• [36107, 3612571607; 7].• [36161, 3616490201; 7].• [36559, 3578646559; 7].• [36671, 3901746671; 7].• [36749, 3554916749; 7].• [36791, 3678150191; 7].• [37549, 1776637549; 7].• [37579, 3785359579; 7].

• [37589, 3757921589; 7].• [37589, 3757921589; 7].• [37699, 3769771459; 7].• [37783, 3778453333; 7].• [38047, 3804776257; 7].• [38047, 3804776257; 7].• [38167, 3816739501; 7].• [38239, 3435548239; 7].• [38351, 3388088351; 7].• [38603, 3819365603; 7].

• [38651, 4967138651; 7].• [38723, 3832076723; 7].• [38767, 3889070767; 7].• [38933, 3612518933; 7].• [39103, 3913714603; 7].• [39107, 3957405107; 7].• [39373, 3937657693; 7].• [39863, 3986374187; 7].• [39989, 3926634989; 7].• [40099, 4355020099; 7].

• [40127, 4011021827; 7].• [40351, 4035179947; 7].• [40543, 4051942243; 7].• [40751, 4286530751; 7].• [40853, 4043695853; 7].• [40883, 4088363987; 7].• [40933, 4093997473; 7].• [41177, 4454801177; 7].• [41221, 4122382261; 7].• [41479, 4145195179; 7].

• [41651, 4117325651; 7].• [41813, 4181498183; 7].• [42397, 4239971677; 7].• [42667, 4236105667; 7].• [42697, 8169042697; 7].• [43207, 4360084207; 7].• [43597, 4353436897; 7].• [43721, 4323415721; 7].• [43969, 4396774489; 7].• [43991, 4393582091; 7].

• [44027, 6571544027; 7].• [44159, 4909664159; 7].• [44179, 4417222729; 7].• [44371, 4139984371; 7].• [44417, 4441559747; 7].• [44623, 4462335217; 7].• [45247, 4522575847; 7].• [45329, 4532995373; 7].• [45341, 4535950841; 7].• [45361, 4586298361; 7].

• [46093, 4602677293; 7].• [46133, 7120546133; 7].• [46183, 5513146183; 7].• [46399, 4632369199; 7].• [46441, 4644352681; 7].• [46523, 4696930523; 7].• [46817, 4688119217; 7].• [46831, 4683214681; 7].• [47111, 5434847111; 7].• [47137, 4733549137; 7].

• [47207, 4722595607; 7].• [47293, 4731581293; 7].• [47389, 4738486759; 7].• [47501, 4757329901; 7].• [47533, 2126747533; 7].• [47569, 4756650919; 7].• [47711, 4377497711; 7].• [47791, 4779256351; 7].• [47797, 1964747797; 7].• [48029, 3355848029; 7].

• [48571, 4857661261; 7].• [48751, 4875394261; 7].• [49003, 3426649003; 7].• [49139, 4918526339; 7].• [49207, 4920528997; 7].• [49369, 4936461919; 7].• [49477, 3396949477; 7].• [49991, 4959346991; 7].• [50101, 5553170101; 7].• [50383, 5038851283; 7].

• [50383, 5038851283; 7].• [50833, 5083086133; 7].• [51539, 5153924339; 7].• [51539, 5153924339; 7].• [51539, 5153924339; 7].• [52067, 7817452067; 7].• [52121, 5212144331; 7].• [52121, 5212144331; 7].

• [53717, 5370866717; 7].• [54293, 5425397993; 7].• [54437, 5443516097; 7].• [55667, 5562145067; 7].• [55819, 5549266819; 7].• [56039, 5630252039; 7].• [56489, 5644795589; 7].• [57191, 7471857191; 7].• [57373, 5738567773; 7].• [57731, 5774700431; 7].

• [57839, 5786777939; 7].• [57847, 9486657847; 7].• [58229, 1419058229; 7].• [58363, 5950198363; 7].• [58477, 5847792733; 7].• [58679, 5869575479; 7].• [59023, 5960906023; 7].• [59341, 4605659341; 7].

• [59351, 5997974351; 7].• [59447, 9122459447; 7].• [60887, 6034938887; 7].• [61031, 1579261031; 7].• [61129, 6204511129; 7].• [61231, 6123634771; 7].• [61343, 6134374787; 7].• [61409, 6130591409; 7].• [61409, 6130591409; 7].• [61483, 6897391483; 7].

• [61933, 6193346947; 7].• [62017, 6889532017; 7].• [62017, 6202554217; 7].• [62207, 6515102207; 7].• [62533, 6253602463; 7].• [62549, 6254969921; 7].• [62617, 6213206617; 7].• [62743, 3177362743; 7].• [62927, 6292945817; 7].• [62971, 6296001571; 7].

65


• [63211, 6765003211; 7].• [63997, 6467043997; 7].• [64081, 6587764081; 7].• [64081, 6587764081; 7].• [64327, 6276334327; 7].• [64853, 6986974853; 7].• [65239, 8610065239; 7].• [65353, 3756065353; 7].• [65563, 6538649563; 7].• [65761, 2641265761; 7].

• [66173, 6617992943; 7].• [66467, 6352776467; 7].• [67339, 6417037339; 7].• [67477, 6787489477; 7].• [67751, 6742579751; 7].• [67943, 6795364343; 7].• [68371, 7379768371; 7].• [68909, 3252968909; 7].• [69061, 6906141349; 7].• [69341, 6931045841; 7].

• [69431, 6923238431; 7].• [69847, 1103769847; 7].• [69959, 6995495519; 7].• [70003, 6659170003; 7].• [70067, 7006155587; 7].• [70529, 4668370529; 7].• [70607, 7783090607; 7].• [70729, 7072667659; 7].• [71167, 7116211807; 7].• [71257, 7230761257; 7].

• [71537, 7009871537; 7].• [71671, 7167681511; 7].• [71777, 8813771777; 7].• [72161, 7244295161; 7].• [72223, 7218066223; 7].• [72739, 7272070339; 7].• [73189, 7373101189; 7].• [73363, 8208373363; 7].• [73529, 7663753529; 7].• [73637, 7362200537; 7].

• [74093, 7490516093; 7].• [74453, 7445636633; 7].• [74861, 7481989061; 7].• [75133, 7515282133; 7].• [75133, 7515282133; 7].• [75367, 7563177367; 7].• [75683, 4236675683; 7].• [76541, 7655326241; 7].• [76829, 7682410469; 7].• [77003, 9704177003; 7].

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• [82457, 6978382457; 7].• [82487, 3794782487; 7].• [82727, 8271117527; 7].• [83089, 8851853089; 7].• [83423, 8342428163; 7].• [83617, 8361996757; 7].• [84449, 8489673449; 7].• [84457, 8494506457; 7].• [84691, 8469481171; 7].• [85453, 6057085453; 7].

• [85621, 8562182437; 7].• [86017, 8684828017; 7].• [86371, 8637120637; 7].• [86477, 8228786477; 7].• [86477, 8228786477; 7].• [86689, 8668950547; 7].• [87383, 8747805383; 7].• [87473, 8743523573; 7].• [88069, 8537428069; 7].• [88681, 8868161167; 7].

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• [91453, 5577691453; 7].• [91909, 9190229509; 7].• [92219, 9756062219; 7].• [92413, 6293792413; 7].• [92957, 9295182287; 7].• [93383, 5663793383; 7].• [93637, 9351897637; 7].• [93703, 9370321927; 7].• [94007, 9405793907; 7].• [94447, 9445781047; 7].

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• [96443, 6231396443; 7].• [96469, 9634755469; 7].• [96487, 9623112487; 7].• [96979, 9142446979; 7].• [97151, 4445797151; 7].• [97373, 9778327373; 7].• [97373, 9778327373; 7].

• [97871, 8245297871; 7].• [98101, 3504398101; 7].• [98317, 9831246637; 7].• [99109, 9951084109; 7].• [99611, 9965459411; 7].• [99661, 9966129007; 7].• [99713, 9972352913; 7].

• [100169, 10016938199; 7].• [100169, 10016938199; 7].• [100559, 10031821559; 7].• [100943, 10093675343; 7].• [101173, 10110089173; 7].• [101467, 10146313417; 7].• [101957, 10195416107; 7].• [101977, 10193609977; 7].• [101977, 10193609977; 7].• [101999, 10196187899; 7].

• [102059, 10209889559; 7].• [102241, 10220025241; 7].• [102251, 10224674651; 7].• [102253, 21723102253; 7].• [102461, 10242167561; 7].• [103001, 10301664101; 7].• [103319, 17449903319; 7].• [103573, 10395229573; 7].• [103951, 16619503951; 7].• [104119, 18862304119; 7].

66


• [104183, 67200104183; 7].• [104789, 10465115789; 7].• [104801, 10497596801; 7].• [105389, 10538745839; 7].• [105761, 97467105761; 7].• [106213, 10732576213; 7].• [106727, 10675876727; 7].• [106727, 10675876727; 7].• [106727, 10675876727; 7].• [106751, 10675668221; 7].

• [106753, 13717306753; 7].• [106957, 12754006957; 7].• [107033, 84075107033; 7].• [108457, 52104108457; 7].• [109379, 80997109379; 7].• [109841, 10984165343; 7].• [109937, 10993727993; 7].• [109961, 10973716961; 7].• [110339, 11075405339; 7].• [110567, 11055791267; 7].

• [110777, 11051255777; 7].• [110917, 11991200917; 7].• [111187, 11154345187; 7].• [111253, 11125358779; 7].• [111697, 11168880097; 7].• [112153, 11855302153; 7].• [112163, 14612212163; 7].• [112589, 13002412589; 7].• [112939, 11293960543; 7].• [113177, 12473213177; 7].

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• [146513, 14657516513; 7].• [146519, 14448566519; 7].• [147487, 14748801577; 7].• [147607, 14710605607; 7].• [147787, 14776739587; 7].• [147853, 12322947853; 7].• [148073, 14802388673; 7].• [148091, 16726648091; 7].• [148147, 14832949147; 7].• [148669, 14867235169; 7].

• [148783, 14839015783; 7].• [149381, 14939320481; 7].• [150053, 85719150053; 7].• [150067, 16929450067; 7].• [150077, 15001111577; 7].• [151157, 15466921157; 7].• [151241, 84000151241; 7].• [151477, 15138883477; 7].• [151523, 96663151523; 7].• [151531, 15723481531; 7].

67


• [151609, 15703531609; 7].• [151651, 13815451651; 7].• [152017, 15201525637; 7].• [152039, 60543152039; 7].• [152287, 15223163287; 7].• [152287, 15223163287; 7].• [152377, 18801452377; 7].• [152393, 15235469693; 7].• [152407, 18135752407; 7].• [152443, 15244521613; 7].

• [152959, 32151152959; 7].• [153427, 15331056427; 7].• [153641, 14740053641; 7].• [153911, 15392403911; 7].• [153911, 15392403911; 7].• [153911, 15392403911; 7].• [153997, 15399764587; 7].• [153997, 15399764587; 7].• [154423, 15617014423; 7].• [154439, 15119524439; 7].

• [154733, 15427945733; 7].• [155251, 14435255251; 7].• [155327, 77814155327; 7].• [155663, 20127155663; 7].• [155699, 15553043699; 7].• [155893, 15541949893; 7].• [156347, 15676284347; 7].• [156719, 15289416719; 7].• [156823, 11416056823; 7].• [156913, 15699178213; 7].

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• [158363, 15853583363; 7].• [158617, 15861125167; 7].• [159739, 15977585539; 7].• [160201, 16766560201; 7].• [160201, 16766560201; 7].• [160403, 17221960403; 7].• [160879, 16087953439; 7].• [160879, 16087953439; 7].• [161237, 16404941237; 7].• [161591, 16157738591; 7].

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• [171467, 17149203767; 7].• [171491, 17339361491; 7].• [171529, 17179536529; 7].• [171653, 95469171653; 7].• [172313, 17231318543; 7].• [172313, 17231318543; 7].• [172399, 17239729489; 7].• [172589, 17254268189; 7].• [172849, 21531172849; 7].• [173291, 17329821851; 7].

• [173501, 39186173501; 7].• [173531, 59535173531; 7].• [173617, 17361398617; 7].• [173617, 17361398617; 7].• [173617, 17361398617; 7].• [173707, 17370981427; 7].• [173891, 34524173891; 7].• [174337, 17562474337; 7].• [174337, 17562474337; 7].• [174457, 17730714457; 7].

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• [180179, 16926180179; 7].• [180179, 16926180179; 7].• [180179, 18012538679; 7].• [180287, 18041553287; 7].• [181717, 18171329707; 7].• [182243, 18224364113; 7].• [182243, 18224364113; 7].• [182443, 18244337569; 7].• [182509, 18518312509; 7].• [182593, 18252704593; 7].

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• [184031, 18407892431; 7].• [184577, 15430084577; 7].• [184609, 18462101509; 7].• [184649, 18469193849; 7].• [184993, 14291284993; 7].• [184997, 15613684997; 7].• [185441, 18999695441; 7].• [185767, 18386675767; 7].• [186163, 18611961163; 7].• [186163, 18611961163; 7].

68


• [186191, 14676786191; 7].• [186211, 17745786211; 7].• [186253, 18739926253; 7].• [186379, 18637157389; 7].• [186551, 14579586551; 7].• [186761, 90129186761; 7].• [187909, 14387287909; 7].• [188483, 80661188483; 7].• [188767, 18878336167; 7].• [188801, 18041888801; 7].

• [189349, 18934969171; 7].• [189401, 18944625101; 7].• [189491, 18815289491; 7].• [189491, 18815289491; 7].• [189817, 18116799817; 7].• [189977, 13367289977; 7].• [190709, 10521190709; 7].• [190979, 19097593169; 7].• [191021, 19966061021; 7].• [191461, 19868261461; 7].

• [191497, 65169191497; 7].• [192553, 19184922553; 7].• [192553, 19361742553; 7].• [192617, 15496092617; 7].• [192637, 19267125637; 7].• [192637, 19267125637; 7].• [192743, 19273390043; 7].• [192979, 19297260769; 7].• [193163, 19810123163; 7].• [193663, 15279793663; 7].

• [193813, 14173093813; 7].• [193951, 19395065251; 7].• [193979, 19326919979; 7].• [194027, 19408380227; 7].• [194521, 19452813181; 7].• [194609, 19474964609; 7].• [194609, 19474964609; 7].• [194981, 19727834981; 7].• [195131, 19513832081; 7].• [195407, 19540835387; 7].

• [195733, 19573826533; 7].• [195733, 19573826533; 7].• [196003, 19600131883; 7].• [196879, 19687815529; 7].• [197059, 19406387059; 7].• [197551, 19752358651; 7].• [197683, 12346097683; 7].• [198031, 93846198031; 7].• [198221, 19822765301; 7].• [198223, 19822348081; 7].

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• [200191, 20646980191; 7].• [200231, 20079476231; 7].• [200237, 20063711237; 7].• [200587, 54405200587; 7].• [200597, 33270200597; 7].• [201287, 97860201287; 7].• [202387, 20934712387; 7].• [202637, 24954502637; 7].• [203057, 29598803057; 7].• [203233, 53964203233; 7].

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• [221251, 22167821251; 7].• [221251, 22167821251; 7].• [221251, 22167821251; 7].• [221413, 22160198413; 7].• [221413, 22141456723; 7].• [222293, 25665522293; 7].• [222317, 22547892317; 7].• [222403, 22247120503; 7].• [223403, 41034223403; 7].• [223711, 22157233711; 7].

• [223747, 22374617407; 7].• [224683, 24761324683; 7].• [225683, 22566267683; 7].• [225683, 22566267683; 7].• [225749, 22320495749; 7].• [225931, 22414785931; 7].• [225943, 22594330987; 7].• [226637, 34194226637; 7].• [226697, 22355026697; 7].• [226697, 22355026697; 7].

• [227377, 29265827377; 7].• [227497, 22749695497; 7].• [227497, 22749695497; 7].• [227497, 22749695497; 7].• [227537, 22794509537; 7].• [228023, 28889628023; 7].• [228197, 26244528197; 7].• [228383, 22143888383; 7].• [228737, 22819095737; 7].• [228853, 22885342483; 7].

69


• [228853, 22885342483; 7].• [228869, 22886319329; 7].• [229153, 22915397233; 7].• [229153, 22915397233; 7].• [229739, 22904089739; 7].• [230257, 23024451757; 7].• [230357, 35424230357; 7].• [230683, 23068888603; 7].• [231269, 23123677769; 7].• [231481, 26842431481; 7].

• [231967, 23196836857; 7].• [232013, 23201560463; 7].• [232381, 23238457861; 7].• [232823, 13419232823; 7].• [233113, 46305233113; 7].• [233267, 24477833267; 7].• [233329, 23332096429; 7].• [233851, 24654533851; 7].• [233993, 23630663993; 7].• [234281, 25277634281; 7].

• [234281, 23428027781; 7].• [234343, 23425125343; 7].• [234529, 23472060529; 7].• [234791, 28142934791; 7].• [234907, 29563134907; 7].• [235369, 14616235369; 7].• [235889, 23589565889; 7].• [235889, 23589565889; 7].• [235889, 23589565889; 7].• [235919, 23591838779; 7].

• [236143, 23614110973; 7].• [236947, 23992736947; 7].• [236947, 23992736947; 7].• [237163, 23758041163; 7].• [237277, 29486337277; 7].• [237619, 80682237619; 7].• [237959, 37968237959; 7].• [238019, 58218238019; 7].• [239027, 23902772213; 7].• [239201, 23915102201; 7].

• [239329, 23438189329; 7].• [239543, 23931617543; 7].• [239713, 23972647213; 7].• [240677, 24586440677; 7].• [240677, 24586440677; 7].• [240859, 57594240859; 7].• [242491, 66318242491; 7].• [242647, 24264127267; 7].• [242773, 24277023973; 7].• [242927, 29207042927; 7].

• [243197, 24800223197; 7].• [243433, 24117633433; 7].• [243829, 24389820229; 7].• [243851, 60018243851; 7].• [243851, 24206733851; 7].• [244313, 24741664313; 7].• [244403, 24442340003; 7].• [245029, 28231745029; 7].• [245173, 24517984993; 7].• [245527, 23810345527; 7].

• [245671, 24556364671; 7].• [245783, 24552710783; 7].• [245941, 24594366331; 7].• [246277, 24623143177; 7].• [246343, 22933146343; 7].• [246469, 24438996469; 7].• [247223, 86235247223; 7].• [247901, 24727210901; 7].• [247997, 24799268957; 7].• [248533, 24663698533; 7].

• [248657, 24862100357; 7].• [248813, 24814997813; 7].• [249211, 24921498391; 7].• [249871, 24987497401; 7].• [250787, 25397020787; 7].• [250853, 58842250853; 7].• [251033, 12561251033; 7].• [251437, 25143724141; 7].• [251663, 25162126163; 7].• [251701, 25178405401; 7].

• [252017, 25273110017; 7].• [252323, 25215060323; 7].• [252449, 25247519249; 7].• [253153, 25315461043; 7].• [253501, 25367833501; 7].• [253501, 25367833501; 7].• [253601, 25367497601; 7].• [253601, 25367497601; 7].• [255173, 26496255173; 7].• [255173, 26496255173; 7].

• [255473, 25547755913; 7].• [255667, 25569814867; 7].• [256211, 26124256211; 7].• [256211, 26124256211; 7].• [256483, 69753256483; 7].• [256661, 25849636661; 7].• [257611, 25723871611; 7].• [257707, 25590527707; 7].• [258143, 11991258143; 7].• [258277, 25847583277; 7].

• [258277, 25825955377; 7].• [259169, 25985302169; 7].• [259183, 25739959183; 7].• [259183, 25739959183; 7].• [260483, 26048244263; 7].• [260587, 25309160587; 7].• [260921, 46002260921; 7].• [261031, 26393481031; 7].• [261799, 26179491379; 7].• [262513, 22814662513; 7].

• [262681, 26748382681; 7].• [262739, 23985562739; 7].• [262747, 23459662747; 7].• [263399, 29307263399; 7].• [263399, 29307263399; 7].• [263429, 26344887329; 7].• [263491, 93891263491; 7].• [263651, 22620863651; 7].• [263863, 26323553863; 7].• [263863, 26323553863; 7].

• [264091, 22413864091; 7].• [264179, 24098664179; 7].• [264949, 28832064949; 7].• [265403, 26541949403; 7].• [265403, 26541949403; 7].• [265547, 26524966547; 7].• [265871, 26587820111; 7].• [266491, 26649113503; 7].• [266599, 26659395559; 7].• [266681, 26507606681; 7].

• [266701, 25395566701; 7].• [266899, 26685328099; 7].• [267317, 26383767317; 7].• [267317, 26383767317; 7].• [267593, 44340267593; 7].• [267611, 26768420411; 7].• [267677, 26798304677; 7].• [268237, 26872078237; 7].• [268237, 26872078237; 7].• [268283, 26545468283; 7].

• [268283, 26545468283; 7].• [268283, 26859187283; 7].• [268733, 25947868733; 7].• [268993, 26853568993; 7].• [268993, 26853568993; 7].• [268993, 26853568993; 7].• [269341, 26934725401; 7].• [269641, 26964823201; 7].• [270209, 27020585279; 7].• [270437, 27033147437; 7].

• [270931, 27856560931; 7].• [271573, 27157321693; 7].• [271573, 27157321693; 7].• [272191, 27217014091; 7].• [272539, 27572432539; 7].• [272737, 55839272737; 7].• [273323, 27332294813; 7].• [274423, 21214474423; 7].• [274709, 27479311109; 7].• [274847, 27288724847; 7].

70


• [275087, 27508721651; 7].• [275459, 27545821229; 7].• [275531, 27497645531; 7].• [275581, 27515168581; 7].• [275767, 27557552767; 7].• [275783, 27110225783; 7].• [275939, 27591177539; 7].• [276011, 27601211531; 7].• [276833, 17418276833; 7].• [276869, 27674994869; 7].

• [277051, 54942277051; 7].• [277483, 27482557483; 7].• [277919, 27791039819; 7].• [278687, 27868153097; 7].• [279479, 24188079479; 7].• [279593, 27954299393; 7].• [279613, 27987246613; 7].• [279659, 27965440289; 7].• [279847, 27984441577; 7].• [280069, 28084207069; 7].

• [280199, 28919590199; 7].• [280339, 28532380339; 7].• [280339, 28532380339; 7].• [280373, 28015315373; 7].• [280583, 28478680583; 7].• [280583, 28478680583; 7].• [280771, 28964740771; 7].• [280949, 22541680949; 7].• [281887, 28173839887; 7].• [282461, 28223391461; 7].

• [282617, 28263781517; 7].• [282713, 28278063113; 7].• [283117, 24123283117; 7].• [283117, 24123283117; 7].• [283133, 28310752133; 7].• [283289, 28461853289; 7].• [283579, 28349659579; 7].• [283793, 28337974793; 7].• [283949, 28668013949; 7].• [284153, 28220174153; 7].

• [284369, 57561284369; 7].• [284407, 28441728307; 7].• [284483, 28448954243; 7].• [284573, 80634284573; 7].• [284593, 28465955593; 7].• [284783, 28472208683; 7].• [284819, 28483273619; 7].• [285161, 20979285161; 7].• [285773, 28577828333; 7].• [285827, 28588974527; 7].

• [285839, 25067085839; 7].• [285977, 28599824777; 7].• [286243, 23758486243; 7].• [286981, 28186696981; 7].• [287669, 49842287669; 7].• [288007, 28845108007; 7].• [288007, 28845108007; 7].• [288181, 28182408181; 7].• [288907, 28893628807; 7].• [288997, 28314168997; 7].

• [289103, 28742989103; 7].• [289103, 28742989103; 7].• [289111, 29785189111; 7].• [289141, 23824789141; 7].• [290317, 23217890317; 7].• [290539, 29052599839; 7].• [290879, 27512390879; 7].• [291103, 29119836103; 7].• [291103, 29119836103; 7].• [291833, 29183693393; 7].

• [291979, 29197793929; 7].• [291983, 29194704083; 7].• [292091, 29957302091; 7].• [292841, 29284659281; 7].• [293941, 29399647141; 7].• [294157, 83244294157; 7].• [294317, 29424906317; 7].• [294347, 29434523927; 7].• [294659, 29465575259; 7].• [294659, 29465575259; 7].

• [294703, 99351294703; 7].• [295727, 29572710977; 7].• [295727, 29572710977; 7].• [296183, 24318296183; 7].• [296183, 24318296183; 7].• [296353, 53466296353; 7].• [297169, 29716533109; 7].• [297233, 29721519533; 7].• [297439, 29767530439; 7].• [297971, 29794921271; 7].

• [298343, 18618298343; 7].• [298559, 29855530799; 7].• [298607, 29439298607; 7].• [298607, 29439298607; 7].• [298607, 29439298607; 7].• [298801, 47229298801; 7].• [299371, 29937742411; 7].• [299807, 29982178307; 7].• [300647, 36862200647; 7].• [300749, 30074861909; 7].

• [300809, 73527300809; 7].• [300851, 30764040851; 7].• [301027, 30102346297; 7].• [301429, 32913601429; 7].• [301619, 30161914259; 7].• [301619, 30161914259; 7].• [301627, 58926301627; 7].• [301703, 30387541703; 7].• [301943, 30874081943; 7].• [302167, 30240590167; 7].

• [302399, 30237915899; 7].• [302663, 35565902663; 7].• [302663, 30257135663; 7].• [302891, 30289193261; 7].• [302891, 30289193261; 7].• [303073, 30615303073; 7].• [303073, 30615303073; 7].• [304049, 31380604049; 7].• [304223, 30422335121; 7].• [304501, 30450150319; 7].

• [304687, 30167344687; 7].• [305101, 30598964101; 7].• [305419, 16611305419; 7].• [305533, 37019705533; 7].• [305861, 30870725861; 7].• [306041, 30645966041; 7].• [306041, 30645966041; 7].• [306517, 30651784861; 7].• [307471, 30771511471; 7].• [308317, 30836531617; 7].

• [308383, 30838715863; 7].• [308641, 34821008641; 7].• [309277, 30928773277; 7].• [309277, 30928773277; 7].• [309521, 59046309521; 7].• [309853, 30130389853; 7].• [309929, 30992725019; 7].• [310823, 31084689323; 7].• [311881, 31183319581; 7].• [311981, 31198146179; 7].

• [312073, 31206813973; 7].• [312253, 31225129963; 7].• [313351, 31334823451; 7].• [313477, 31347715351; 7].• [313589, 31168753589; 7].• [313639, 31363771429; 7].• [313721, 31364926721; 7].• [314423, 31497836423; 7].• [314543, 31634204543; 7].• [314779, 31475808379; 7].

• [315223, 31554831223; 7].• [315461, 51432315461; 7].• [316429, 31641078229; 7].• [316493, 31673230493; 7].• [317197, 31719765811; 7].• [317483, 31951817483; 7].• [317483, 31951817483; 7].• [318473, 31873719473; 7].• [318629, 31862835509; 7].• [319339, 31937075239; 7].

71


• [319733, 31971454733; 7].• [319733, 31971454733; 7].• [319747, 31974437017; 7].• [320081, 32008349141; 7].• [320107, 32050238107; 7].• [320153, 32082044153; 7].• [321199, 33126321199; 7].• [321199, 33126321199; 7].• [321247, 33537321247; 7].• [321247, 33537321247; 7].

• [321469, 32660931469; 7].• [322001, 32241289001; 7].• [322571, 32290930571; 7].• [322633, 36716722633; 7].• [322969, 32371852969; 7].• [323377, 32329886377; 7].• [324031, 10854324031; 7].• [324101, 32722044101; 7].• [324293, 32156154293; 7].• [324299, 39198324299; 7].

• [324299, 39198324299; 7].• [324299, 32429665439; 7].• [324529, 86142324529; 7].• [325027, 32503736527; 7].• [325051, 32559541051; 7].• [325723, 86922325723; 7].• [325753, 32571202753; 7].• [325753, 32571202753; 7].• [326023, 32602217023; 7].• [326023, 32602217023; 7].

• [326561, 69993326561; 7].• [327133, 16296327133; 7].• [327419, 32741197109; 7].• [327511, 32751147277; 7].• [327517, 32775867517; 7].

• [327517, 32775867517; 7].• [327529, 32756741929; 7].• [327557, 32710095557; 7].• [327647, 32764992047; 7].• [327647, 32764992047; 7].

• [327721, 32777024821; 7].• [327881, 37775127881; 7].• [328781, 32878749131; 7].• [328931, 32893348601; 7].• [329393, 32939326691; 7].• [329587, 32958330817; 7].• [329683, 32968492423; 7].• [329789, 32978685899; 7].• [330587, 33361680587; 7].• [331207, 33120103207; 7].

• [331207, 33120103207; 7].• [331391, 33136006691; 7].• [331907, 33897151907; 7].• [332251, 33222735451; 7].• [332851, 34434032851; 7].• [332873, 33222227873; 7].• [332881, 33245357881; 7].• [333433, 34199133433; 7].• [333721, 33372611581; 7].• [334157, 33415437017; 7].

• [334987, 33450373987; 7].• [335857, 33742715857; 7].• [336239, 33623961509; 7].• [336239, 33623961509; 7].• [336793, 33671646193; 7].• [337189, 33397477189; 7].• [337219, 33796456219; 7].• [337661, 33720673661; 7].• [338267, 33823924367; 7].• [338731, 33557438731; 7].

• [338731, 33557438731; 7].• [338839, 71388338839; 7].• [339151, 33911458351; 7].• [339151, 33917933551; 7].• [339173, 39033039173; 7].

• [339289, 38729439289; 7].• [339331, 33957063331; 7].• [339373, 33935643373; 7].• [339373, 33935643373; 7].• [340519, 34051680529; 7].

• [340633, 34527130633; 7].• [340777, 34078823677; 7].• [340789, 34078968817; 7].• [340819, 48048340819; 7].• [340931, 12243340931; 7].• [340931, 43608340931; 7].• [341323, 22533341323; 7].• [341851, 34184584051; 7].• [342257, 34225437437; 7].• [342413, 34275138413; 7].• [343639, 37227343639; 7].• [343639, 37227343639; 7].

• [343639, 34393807639; 7].• [343891, 34387455391; 7].• [344273, 34982414273; 7].• [344819, 34481879189; 7].• [344963, 34496965163; 7].• [344963, 34496965163; 7].• [345463, 34594716463; 7].• [345997, 34599551617; 7].• [346391, 34620400391; 7].• [346607, 37420246607; 7].

• [346649, 34663366349; 7].• [346849, 31746946849; 7].• [347629, 32970347629; 7].• [347629, 32970347629; 7].• [347651, 34784663651; 7].

• [347759, 34775277599; 7].• [348269, 35666448269; 7].• [348827, 34843149827; 7].• [349039, 37132249039; 7].• [349663, 34962375163; 7].

• [349759, 34770859759; 7].• [349931, 93009349931; 7].• [350179, 36204350179; 7].• [350179, 36204350179; 7].• [350521, 35052221011; 7].• [350887, 34854050887; 7].• [350947, 38569250947; 7].• [351041, 35104551641; 7].• [351041, 35104551641; 7].• [351059, 35107590359; 7].

• [351217, 35121605227; 7].• [351287, 35128817387; 7].• [351287, 35128817387; 7].• [351653, 35180643653; 7].• [351779, 35943531779; 7].• [352349, 35231044349; 7].• [352349, 35231044349; 7].• [352361, 30906352361; 7].• [352741, 35277784441; 7].• [353333, 35333837453; 7].

• [353389, 35360954389; 7].• [353501, 97017353501; 7].• [353783, 37272653783; 7].• [354149, 35414888759; 7].• [354391, 35439634891; 7].• [354391, 35439634891; 7].• [354619, 35461897459; 7].• [354727, 35497878727; 7].• [354973, 35497428193; 7].• [355361, 35538110561; 7].• [355819, 35588211319; 7].

72


5.6 Six Digits RepetitionsBelow are some examples of representations:

• [83, 83217077; 7] I 8383 21707783 217077 21707783 217077 217077 21707783 217077 217077 217077 21707783 217077 217077 217077 217077 21707783 217077 217077 217077 217077 217077 217077.

• [97, 67168597; 7]. I 97671685 97671685 671685 97671685 671685 671685 97671685 671685 671685 671685 97671685 671685 671685 671685 671685 97671685 671685 671685 671685 671685 671685 97.


• [131, 137070701; 7].• [167, 179831567; 7].• [229, 371406229; 7].• [311, 383777411; 7].• [439, 159495439; 7].• [523, 572576023; 7].• [683, 658396883; 7].• [719, 575526719; 7].• [773, 779429633; 7].• [911, 914514791; 7].

• [1009, 1002746599; 7].• [1103, 1129353803; 7].• [1259, 1170184259; 7].• [1307, 1923034307; 7].• [1319, 1843549319; 7].• [1453, 1454592493; 7].• [1607, 1622428007; 7].• [1693, 1625502493; 7].

• [1949, 1948498709; 7].• [2243, 1565132243; 7].• [2243, 2222736743; 7].• [2273, 2276145863; 7].• [2357, 6548222357; 7].• [2371, 2295134371; 7].• [2371, 2377016101; 7].• [2447, 2447238749; 7].

• [2593, 2591448373; 7].• [2699, 5675672699; 7].• [2719, 2711920879; 7].• [2797, 4494632797; 7].• [2969, 2877905969; 7].• [3061, 2527143061; 7].• [3217, 3294607117; 7].• [3229, 3480984229; 7].

• [3299, 3291384749; 7].• [3739, 3539721739; 7].• [3803, 3863793803; 7].• [3803, 3863793803; 7].• [3803, 3863793803; 7].• [3847, 3848538607; 7].• [3923, 3926077193; 7].• [4363, 4311480763; 7].

• [4421, 2971924421; 7].• [4733, 1620154733; 7].• [4759, 4754457889; 7].• [5051, 5037648851; 7].• [5171, 8956925171; 7].• [5197, 5197628383; 7].• [5693, 5692499843; 7].• [5861, 5187866861; 7].

• [6133, 6164906833; 7].• [6803, 6149100803; 7].• [7307, 6091897307; 7].• [7591, 7536353191; 7].• [7669, 7646788069; 7].• [7727, 7712049827; 7].• [7841, 7845058691; 7].• [7879, 7110712879; 7].

• [7907, 7903428677; 7].• [8089, 8956592089; 7].• [8123, 8124631133; 7].• [8287, 8287057057; 7].• [8669, 8668392869; 7].• [8669, 8668392869; 7].• [8689, 2564318689; 7].• [8783, 8783562317; 7].

73


• [8933, 6661208933; 7].• [8971, 8763035971; 7].• [9587, 4831899587; 7].• [9689, 1078149689; 7].• [9817, 4695399817; 7].• [9887, 9142359887; 7].• [9887, 9142359887; 7].

• [10099, 10685503099; 7].• [10181, 10517965181; 7].• [10223, 10276704623; 7].• [10343, 15427870343; 7].• [10477, 10415829877; 7].• [10501, 13982440501; 7].• [10613, 10618540073; 7].• [10709, 10717117109; 7].

• [11243, 14788001243; 7].• [11287, 17612501287; 7].• [11483, 79808411483; 7].• [11497, 11970137497; 7].• [11939, 11939606123; 7].• [12073, 12077612083; 7].• [12421, 45639312421; 7].• [12473, 12422367173; 7].

• [12553, 65165112553; 7].• [12601, 12608273371; 7].• [12613, 26010612613; 7].• [12647, 12641046857; 7].• [12781, 12735147781; 7].• [12781, 12735147781; 7].• [12853, 12855937753; 7].• [12853, 12855937753; 7].

• [12911, 12913742201; 7].• [12941, 12911554241; 7].• [13037, 17703643037; 7].• [13159, 13195195159; 7].• [13159, 13195195159; 7].• [13597, 28043413597; 7].• [13757, 13752680867; 7].• [14087, 12169514087; 7].

• [14087, 12169514087; 7].• [14249, 15642914249; 7].• [14249, 15642914249; 7].• [14251, 22383914251; 7].• [14341, 12686744341; 7].• [14407, 73642814407; 7].• [14563, 14678678563; 7].• [14621, 12755624621; 7].

• [14759, 14758400849; 7].• [15077, 15439950077; 7].• [15121, 15124189501; 7].• [15173, 15173682269; 7].• [15241, 15243352231; 7].• [15359, 15335175059; 7].• [15401, 15851592401; 7].• [15649, 15945945649; 7].

• [15649, 15945945649; 7].• [15971, 19760595971; 7].• [16103, 16108071143; 7].• [16223, 19087546223; 7].• [16319, 16319300951; 7].• [16519, 16544828719; 7].• [16573, 16549320673; 7].• [16823, 16822240913; 7].

• [16903, 16951397503; 7].• [16979, 16296415979; 7].• [17021, 17017224221; 7].• [17093, 17093423717; 7].• [17159, 69524717159; 7].• [17341, 17708897341; 7].• [17341, 17708897341; 7].• [17383, 12686957383; 7].

• [17383, 17855876383; 7].• [17623, 17626685983; 7].• [17791, 11752877791; 7].• [17807, 17260757807; 7].• [17807, 17260757807; 7].• [17863, 91167317863; 7].• [18311, 18144795311; 7].• [18583, 18585621073; 7].

• [18701, 18944349701; 7].• [18787, 18742163887; 7].• [18911, 18635418911; 7].• [18911, 18635418911; 7].• [18911, 18635418911; 7].• [19069, 18509629069; 7].• [19231, 18646759231; 7].• [19489, 18155779489; 7].

• [19559, 19346983559; 7].• [19577, 31697419577; 7].• [20071, 20482517071; 7].• [20261, 20918666261; 7].• [20333, 26305250333; 7].• [20599, 20569493299; 7].• [20903, 20909045123; 7].• [21059, 59770221059; 7].

• [21187, 21185793697; 7].• [21211, 74604621211; 7].• [21419, 29904021419; 7].• [21419, 29904021419; 7].• [21521, 21959511521; 7].• [21521, 21959511521; 7].• [21559, 24613701559; 7].• [21601, 21345324601; 7].

• [21817, 39662721817; 7].• [21859, 29296071859; 7].• [21871, 23419221871; 7].• [21871, 23419221871; 7].• [21991, 21220038991; 7].• [22013, 22361074013; 7].• [22031, 25262602031; 7].• [22039, 22034422609; 7].

• [22271, 22731094271; 7].• [22271, 22270602071; 7].• [22391, 22354163291; 7].• [22651, 22651853503; 7].• [22717, 29482342717; 7].• [22721, 22725289901; 7].• [23063, 23226044063; 7].• [23159, 23159125349; 7].

• [23159, 23159125349; 7].• [23251, 23253652111; 7].• [23539, 21288983539; 7].• [23929, 23960406529; 7].• [24083, 77462724083; 7].• [24109, 24126911509; 7].• [24137, 24289212137; 7].• [24151, 24156105331; 7].

• [24169, 24140897569; 7].• [24203, 24204709253; 7].• [24223, 24265868623; 7].• [24379, 32323224379; 7].• [24709, 24774289609; 7].• [24907, 24903320947; 7].• [24979, 24982099579; 7].• [25153, 25151274913; 7].

• [25219, 28175095219; 7].• [25243, 10101025243; 7].• [25343, 25344886703; 7].• [25367, 25366443437; 7].• [25391, 25307839391; 7].• [25423, 28168165423; 7].• [25633, 22610305633; 7].• [25889, 27577645889; 7].

• [25969, 25968329659; 7].• [26083, 24111806083; 7].• [26119, 76986026119; 7].• [26263, 26256433363; 7].• [26357, 57017126357; 7].• [26539, 29534426539; 7].• [26539, 25354796539; 7].• [26539, 29534426539; 7].

• [26561, 26693651561; 7].• [27011, 27019273811; 7].• [27011, 27019273811; 7].• [27551, 39284727551; 7].• [27551, 27707931551; 7].• [28211, 28518742211; 7].• [28789, 28786528489; 7].• [28789, 28786528489; 7].

74


• [28837, 21000238837; 7].• [28933, 28916566933; 7].• [28933, 28916566933; 7].• [28949, 28330603949; 7].• [29009, 26231969009; 7].• [29221, 29227004761; 7].• [29251, 29252929081; 7].• [29473, 29474714923; 7].• [29483, 29283437483; 7].• [29587, 29592792787; 7].

• [29669, 29979209669; 7].• [29669, 29979209669; 7].• [29741, 29722409141; 7].• [30161, 24832530161; 7].• [30341, 34825380341; 7].• [30689, 30661891289; 7].• [30937, 30938624287; 7].• [30949, 30979980649; 7].• [31159, 31159884751; 7].

• [31327, 31232617327; 7].• [31391, 31391696423; 7].• [31481, 31580482481; 7].• [31657, 83447731657; 7].• [31751, 31760872751; 7].• [31751, 31760872751; 7].• [31771, 31776029731; 7].• [31963, 31969941613; 7].• [32077, 96474032077; 7].

• [32309, 78424532309; 7].• [32321, 16350632321; 7].• [32411, 32433818411; 7].• [32411, 32433818411; 7].• [32537, 32142170537; 7].• [32801, 32836174601; 7].• [32801, 32806488791; 7].• [32869, 32457149869; 7].• [32939, 47655332939; 7].• [33107, 33106132427; 7].

AcknowledgementThe author is thankful to T.J. Eckman, Georgia, USA (email: [email protected]) in programming the script to developthis paper.

References[1] H.E. DUDENEY, Amusements in Mathematics, EBD E-Books Directory.com, 1917.

[2] H. HEINZ, Prime Patterns, http://www.magic-squares.net/primes.htm.

[3] G. L. HONAKER, JR. and C. CALDWELL, Prime Curios, 1709,https://primes.utm.edu/curios/page.php/1709.html

[4] J.S. MADACHY, Mathematics on Vacations, Charlers Scriber’s Son, New York, 1966.

[5] C.A. PICKOVER, A Paisson of Mathematics, John Wiley & Sons, Inc., New Jersey, 2005.

[6] I.J. TANEJA, Crazy Representations of Natural Numbers, Selfie Numbers, Fibonacci Sequence,and Selfie Fractions, RGMIA Research Report Collection, 19(2016), Article 179, pp.1-37,http://rgmia.org/papers/v19/v19a179.pdf.

[7] I.J. TANEJA, Hardy Ramanujan Number - 1929, RGMIA Research Report Collection, 20(2017), Article 6,pp.1-50, http://rgmia.org/papers/v20/v20a06.pdf.

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