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: ijtaneja@gmail.com; Web-site: www.numbersmagic.wordpress.com.

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.

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.

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.

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.

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.

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.

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.

• [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.

• [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.

• [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].

• [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].

• [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].

• [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].

• [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.

• [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].

• [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].

• [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].

• [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].

• [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].

• [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].

• [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].

• [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.

• [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].

• [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].

• [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].

• [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].

• [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].

• [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].

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].

• [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.

• [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].

• [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].

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].

• [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].

• [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].

• [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].

• [146917, 14366917; 7].• [175709, 17575709; 7].• [175709, 17575709; 7].• [175709, 17575709; 7].

• [175709, 17575709; 7].• [177101, 17710841; 7].• [183523, 60183523; 7].• [194527, 19664527; 7].

• [199873, 84199873; 7].• [205619, 20725619; 7].• [216023, 23316023; 7].• [220291, 22021591; 7].

• [222659, 22284659; 7].• [235441, 23544181; 7].• [235441, 23544181; 7].• [266599, 69266599; 7].

• [268783, 23368783; 7].• [272341, 27422341; 7].• [280009, 99280009; 7].• [281621, 28162331; 7].

• [282287, 28257287; 7].• [288389, 28838699; 7].• [298087, 60298087; 7].• [299063, 84299063; 7].

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• [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].

• [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].

• [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].

• [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].

• [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].

• [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].

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• [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].

• [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].

• [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].

• [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].

• [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].

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• [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].

• [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].

• [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].

• [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].

• [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].

• [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].

• [77141, 7547897141; 7].• [77269, 5123477269; 7].• [77929, 8926877929; 7].• [78007, 7250808007; 7].• [78779, 7877963507; 7].• [78803, 7880354717; 7].• [79229, 7839169229; 7].• [79279, 7954114279; 7].• [79433, 7943624903; 7].• [79589, 7934257589; 7].

• [79589, 7958683529; 7].• [80239, 8023405009; 7].• [80317, 9447980317; 7].• [80329, 8036057329; 7].• [80329, 8036057329; 7].• [80347, 8032945747; 7].• [80387, 8038345247; 7].• [81223, 8122612543; 7].• [82067, 8229874067; 7].• [82171, 8271898171; 7].

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• [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].

• [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].

• [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].

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].

• [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].

• [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: jeek@jeek.net) 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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