all the prime factors of the smarandache concatenated numbers upto the first not factor
TRANSCRIPT
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 1/32
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 2/32
NFSNET - Number Field Sieve GGNFS - A Number Field Sieve implementationGMP-ECM 6.0.1
Messages
[ May 20, 2008 ]Greg Childers (email) factorized Sm99 ! [ go to entry ]
Hi Patrick,
I finally got around to factoring Sm99 by SNFS. As for Sm94, I usedthe GGNFS lattice siever and msieve for the postprocessing. The factors are
P65: 37726668883887938032416757819314355053940153680075342644295667759
P107: 14627910783072606795565990651314126145674770336615677946549896262532933945988541999815567058347827465728809
Greg
[ December 9, 2007 ]Greg Childers (email) factorized Sm94 ! [ go to entry ]
Hi Patrick,
Here are the factors of Sm94. This was completed using SNFS.
GGNFS was used for the sieving and msieve for the post-processing.
p79: 1825097233762709447432521941926649289213154260264910537140594516431173070300371
p82: 2674525573684858697560701870658348933916102325593721165422426453989766526938215889
Greg
[ June 8, 2006 ]Sean A. Irvine (email)
Excellent work! (reg. Sm98 by Ph. Strohl)
Sm94 is still struggling. My yield has dropped below 1 and I've sievedto special-q 70M already, but still don't have enough relations.
S.
[ June 6, 2006 ]Philippe Strohl (email)
Hi Patrick !I have factorized the remaining part of the 98th Smarandacheconcatenated number. It was a 126 digits composite.I have broken it with ggnfs (a wonderful program : I have done it witha common laptop computer with "only" 512 megs of ram and a 1.4 GHzceleron M in two weeks).
the results :N = 709891330215674922888729762564179876071621230485156504316706784486089628388958197669654729613064737223993515869067455247541707 ( 126 digits)Divisors found:r1=3588472635471667861938967869443938442910813342994227048889 (pp58)r2=197825482406769698151783117995020967519766027202915861687264259155363 (pp69)
Version: GGNFS-0.77.1-20060513-pentium-m(prp verified prime with apr-cl)
Thanks for your site and for keeping tracks of our work.Best regards.
Philippe Strohl
[ September 11, 2005 ]Sean A. Irvine (email)
The next two have finished:
Sm87 C145 =
Page 2 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not factor ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 3/32
(p51) * (p95)by SNFS, 14 daysSm88 C153 =(p42) * (p51) * (p61)by GNFS, 8 hours
Sm90 looks like it will have to be SNFS since ECM has failedto find a factor.
Regards,Sean.
[ August 29, 2005 ]Philippe Strohl (email)
Hi Patrick !Thanks for accepting my "colouring" idea ! I am very pleased !
I should report you this 39 digits factor for Sm98 (Partial factorization):
Sm98* 2.3^2.23.37.199p16: 1495444452918817(MF)c165: 270825497607069872452415496119443135107702791840293286471110488510476827439126669519712057435717362779439193614301623544632857479569035194034142023605896434694145167
Line=16/32 Curves=47/1000 B1=1000000 factors=0C165 Using B1=1000000, B2=839549780, polynomial Dickson(6), sigma=594918519Step 1 took 28422msStep 2 took 18926ms********** Factor found in step 2: 381502754125464943168932369122248696781Found probable prime factor of 39 digits:381502754125464943168932369122248696781
Composite cofactor709891330215674922888729762564179876071621230485156504316706784486089628388958197669654729613064737223993515869067455247541707 has 126 digits
Thanks again.Best regards.Philippe Strohl
[ August 29, 2005 ]Sean A. Irvine (email)
Here is the last part of Sm86:
10828687641092318839822035841363590407263202742239027773 (p56) *1089075252400674157091531724111232381528208779232955680665273 (p61)by GNFS, 2 days.
As before I'm now working on Sm87.
Sean.
[ August 28, 2005 ]Sean A. Irvine (email) latest results.
Here are my latest results. Like I mentioned earlierI expect to complete Sm86 today as well:
Sm83 C134
21875480270521598141087357354188092945840550359281483 (p53) *3966169790267211790412249283896602109358687165012835285295541472324348526743126307 (p82)by SNFS, 8 days
Sm85 C158 =120549814855596987772827562271063563633851059 (p45) *Using B1=11000000, sigma=17081242912112809210944968177871685727287164545437750155430310661 (p55) *197843626412162026434764405036310959588059884460495810550047 (p60)by GNFS, 1 day
Sm86 C154 =718252229986396496762902999331863301257 (p39) * C116Using B1=11000000, sigma=3414478964C116 by GNFS nearly done
Page 3 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not factor ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 4/32
Sm87 C145 Sieving by SNFS started 2005-08-29.
Others with B1=1e6 (I have now completed 1000 curves with B1=1e6on all Sm(n), n <= 200)
Sm114 8678622406220213516465050301044327Sm159 45941358846148651407783221723920871719Sm171 40202471819457246557501649563881337Sm193 5167315927941164272437909427556797
Sean.
It seems that a fierce competition is going on. But to avoidduplicate work and loose valuable cpu time I advice stronglyto make arrangements among yourselves!
[ August 28, 2005 ]Philippe Strohl (email)
..., for example : the smallest unfactored Sm number is sm83(unfactored) but I have factored last year Sm85 (involving a p45 found byecm) and Sm86 (with a p39 and a ggnfs on the remaining c115)... Thisrepresents quite a large amount of cpu work. Sm84 factorization is stillknown on M Fleuren page...
[ August 3, 2005 ]Sean A. Irvine (email) completely factorized Sm78 ! [ go to entry ]
Sm78 C139 =205155431830422787082756234197593935249202704547671264423 (p57) *17403902113720391120287411398887911225298966708915583006414519403038472992542973083 (p83)by GNFS (General Number Field Sieve), 9 days
Here is a bunch more factors for higher values which I have not seenpreviously reported. All these were found with ECM B1=1e6.
Sm89 496118159817126721484175235476073Sm89 26459905787227421825352754831024262009257.P64Sm92 46731404628893905607210235741707Sm93 19544056951015647623992763251 Sm95 244987542265129586458446183157595351.P141Sm100 970447246795177523033247400823.P118Sm106 95383501607400293616004374931Sm106 54259599094002572583355411045946413 Sm108 132761751746390611923240080737166083.P161Sm109 9943216978062352390003139833531Sm114 2042059881000388200555074336219 Sm116 9787002048140152171263515060558503699.P198Sm121 105299178204417486675841093021769.P214Sm123 12347002211187670552593982429Sm123 2829927788416784955921382453753Sm125 295999706346724665505289Sm137 144065103514544138702103468451 Sm148 8817212782626223819399721069204897.P254Sm152 4103096315830350734534473515557Sm152 12805089500421274253268517941967Sm152 17815076027044127272632744936161.P205Sm154 32063206397901252963254536935569Sm159 11855111297257593607972759339201Sm160 64603936118676024484144135734907Sm162 22260247937572504750086047Sm164 1039418554780603268384723777072953 Sm165 13183356310254866666237435750357.P328Sm176 1011379313630785579015894871Sm183 553245689211853052761209813199 Sm184 677008100402429325901609057.P342Sm187 1080829169904060835770214147747.P411Sm193 419908232491384495189 Sm195 165897663095213559529993681.P412Sm198 14158849264684185910199571953
Further, after studying Backstrom's work on Rsm76 I am now able to generateSNFS polynomials for all the remaining Sm numbers below 100. It would havebeen much faster to do Sm78 by SNFS, but I had already started it beforeworking out how to apply SNFS to the number. It should be possible to completeall values up to Sm(100) by SNFS, although a few will be quite difficult runs.
The next smallest unfactored number of this form is now Sm83 [ C134 ].
Sean.
Page 4 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not factor ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 5/32
[ November 18, 2004 ]Sean A. Irvine (email) completely factorized Sm75 ! [ go to entry ]
It took him 13 days, by using GNFS.Well done, congratulations for factoring Sm75(c133) into this p47 * p87 :38824496309870038690197243565592769246963314017 (p47) *219358378032318168161320006998916878634145966511629131235131312083699783021949850982403 (p87)
Next challenge is this composite factor of 139 digits of Sm78 :3570505053674714753162296261527331568459971771942/ 9181309659088118527251315326728064046015264067596/ 03889145976969679985423963150530264526109
[ March 23, 2004 ]Philippe Strohl (email) completely factorized Sm73 ! [ go to entry ]
Hello Patrick !
This mail to inform you that the factorization of the 73th concatenated smarandache number is now completewith the discovery of a p46 by GMP-ECM...
Sm73 = 37907.p46.p87p46: 1612352371081094864112011094480307952600705089p87: 201992666185187831800817490810938117880341395186600971262233773863756955874363353778851...Sm74 factorization is known and the next composite to challenge is Sm75with no factors expected below 35 digits...
Sm74 = 2.3.7.1788313.21565573.p20.p25.p31.p49p20: 99014155049267797799p25: 1634187291640507800518363(PZ)p31: 1981231397449722872290863561307p49: 2377534541508613492655260491688014802698908815817
Sm75* 3.5^2.193283.c133c133:851647853845481367839983983361331811035304896846801931077529055832/ 3936344974946612980172082837107906069172212808249295700548030242851
The List of Sm Factors
1
1
12
22
3
123
341
1234
2
617
Sm1 = p1 = unity
1
Sm2 = (p1)^2 * p1
2^2 *
3
Sm3 = p1 * p2 = semiprime
3 *
41
Sm4 = p1 * p3 = semiprime
2 *
Page 5 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not factor ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 6/32
12345
3
5
823
123456
26
3
643
1234567
127
9.721
12345678
2
32
47
14.593
123456789
32
3.607
3.803
12345678910
2
5
1.234.567.891
1234567891011
3
7
13
67
107
630.803
617
Sm5 = p1 * p1 * p3
3 *
5 *
823
Sm6 = (p1)^6 * p1 * p3
2^6 *
3 *
643
Sm7 = p3 * p4 = semiprime
127 *
9721
Sm8 = p1 * (p1)^2 * p2 * p5
2 *
3^2 *
47 *
14593
Sm9 = (p1)^2 * p4 * p4
3^2 *
3607 *
3803
Sm10 = p1 * p1 * p102 *
5 *
1234567891
Page 6 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not factor ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 7/32
1234567...12
23
32.437
2.110.805.449
1234567...13
113
125.693
869.211.457
1234567...14
2
3
205.761.315.168.520.219
1234567...15
3
58.230.452.606.740.808.761
1234567...16
22
2.507.191.691
1.231.026.625.769
1234567...17
32
47
4.993
584.538.396.786.764.503
Sm11 = p1 * p1 * p2 * p2 * p3 * p6
3 *
7 *
13 *
67 *
107 *
630803
Sm12 = (p1)^3 * p1 * p4 * p10
2^3 *
3 *
2437 *
2110805449
Sm13 = p3 * p6 * p9
113 *125693 *
869211457
Sm14 = p1 * p1 * p18
2 *
3 *
205761315168520219
Sm15 = p1 * p1 * p19
3 *
5 *
8230452606740808761
Sm16 = (p1)^2 * p10 * p13
2^2 *
2507191691 *
1231026625769
Sm17 = (p1)^2 * p2 * p4 * p18
3^2 *
47 *
4993 *
584538396786764503
Page 7 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not factor ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 8/32
1234567...18
2
32
97
88.241
801.309.546.900.123.763
1234567...19
13
43
79
281
1.193
833.929.457.045.867.563
1234567...20
25
3
5
323.339
3.347.983
2.375.923.237.887.317
1234567...21
3
17
37
43
103
131
140.453
802.851.238.177.109.689
1234567...22
2
7
1.427
3.169
85.829
2.271.991.367.799.686.681.549
Sm18 = p1 * (p1)^2 * p2 * p5 * p18
2 *
3^2 *97 *
88241 *
801309546900123763
Sm19 = p2 * p2 * p2 * p3 * p4 * p18
13 *
43 *
79 *
281 *
1193 *
833929457045867563
Sm20 = (p1)^5 * p1 * p1 * p6 * p7 * p16
2^5 *
3 *
5 *
323339 *3347983 *
2375923237887317
Sm21 = p1 * p2 * p2 * p2 * p3 * p3 * p6 * p18
3 *
17 *37 *
43 *
103 *
131 *
140453 *
802851238177109689
Page 8 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not factor ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 9/32
1234567...23
341
769
13.052.194.181.136.110.820.214.375.991.629
1234567...24
22
3
7
978.770.977.394.515.241
1.501.601.205.715.706.321
1234567...25
52
15.461
31.309.647.077
1.020.138.683.879.280.489.689.401
1234567...26
2
34
21.347
2.345.807
982.658.598.563
154.870.313.069.150.249
1234567...27
33
192
4.547
68.891
40.434.918.154.163.992.944.412.000.742.833
Sm22 = p1 * p1 * p4 * p4 * p5 * p22
2 *
7 *
1427 *
3169 *
85829 *
2271991367799686681549
Sm23 = p1 * p2 * p3 * p32
3 *
41 *
769 *
13052194181136110820214375991629
Sm24 = (p1)^2 * p1 * p1 * p18 * p19
2^2 *
3 *
7 *
978770977394515241 *
1501601205715706321
Sm25 = (p1)^2 * p5 * p11 * p25
5^2 *15461 *
31309647077 *
1020138683879280489689401
Sm26 = p1 * (p1)^4 * p5 * p7 * p12 * p18
2 *
3^4 *
21347 *
2345807 *982658598563 *
154870313069150249
Sm27 = (p1)^3 * (p2)^2 * p4 * p5 * p32
3^3 *
Page 9 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not factor ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 10/32
1234567...28
23
47
409
416.603.295.903.037
192.699.737.522.238.137.890.605.091
1234567...29
3
859
24.526.282.862.310.130.729
19.532.994.432.886.141.889.218.213
1234567...30
2
3
5
13
49.269.439
370.677.592.383.442.753
17.333.107.067.824.345.178.861
1234567...31
29
2.597.152.967
163.915.283.880.121.143.989.433.769.727.058.554.332.
117
1234567...32
22
3
7
45.068.391.478.912.519.182.079
326.109.637.274.901.966.196.516.045.637
19^2 *
4547 *
68891 *
40434918154163992944412000742833
Sm28 = (p1)^3 * p2 * p3 * p15 * p27
2^3 *
47 *
409 *
416603295903037 *
192699737522238137890605091
Sm29 = p1 * p3 * p20 * p26
3 *859 *
24526282862310130729 *
19532994432886141889218213
Sm30 = p1 * p1 * p1 * p2 * p8 * p18 * p23
2 *
3 *
5 *13 *
49269439 *
370677592383442753 *
17333107067824345178861
Sm31 = p2 * p10 * p42
29 *
2597152967 *
163915283880121143989433769727058554332117
Sm32 = (p1)^2 * p1 * p1 * p23 * p30
2^2 *
3 *
7 *
45068391478912519182079 *
326109637274901966196516045637
Page 10 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 11/32
1234567...33
3
23
269
7.547
116.620.853.190.351.161
7.557.237.004.029.029.700.530.634.132.859
1234567...34
2
6.172.839.455.055.606.570.758.085.909.601.061.116.
212.631.364.146.515.661.667
1234567...35
32
5
139
151
64.279.903
4.462.548.227
4.556.722.495.899.317.991.381.926.119.681.186.927
1234567...36
24
32
103
211
39.448.709.943.503.776.711.542.648.338.171.477.043.
440.283.875.433.388.943
1234567...37 71
12.379
4.616.929
3.042.410.911.077.206.144.807.069.396.988.766.146.
557.218.727.107.817
1234567...38
Sm33 = p1 * p2 * p3 * p4 * p18 * p31
3 *23 *
269 *
7547 *
116620853190351161 *
7557237004029029700530634132859
Sm34 = p1 * p58 = semiprime
2 *
6172839455055606570758085909601061116212631364146515661667
Sm35 = (p1)^2 * p1 * p3 * p3 * p8 * p10 * p37
3^2 *
5 *
139 *
151 *
64279903 *
4462548227 *
4556722495899317991381926119681186927
Sm36 = (p1)^4 * (p1)^2 * p3 * p3 * p56
2^4 *
3^2 *
103 *
211 *
39448709943503776711542648338171477043440283875433388943
Sm37 = p2 * p5 * p7 * p52
71 *
12379 *
4616929 *
3042410911077206144807069396988766146557218727107817
Page 11 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 12/32
2
3
86.893.956.354.189.878.775.643
2.367.958.875.411.463.048.104.007.458.352.976.869.124.
861
1234567...39
3
67
311
1.039
6.216.157.781.332.031.799.688.469
305.788.363.093.026.251.381.516.836.994.235.539
1234567...40
22
5
3.169
60.757
579.779
4.362.289.433
79.501.124.416.220.680.469
15.944.694.111.943.672.435.829.023
1234567...41
3
487
493.127
32.002.651
53.545.135.784.961.981.058.419.604.998.638.516.483.
529.257.158.438.201.753
1234567...42
2
3
127
421
22.555.732.187
4.562.371.492.227.327.125.110.177
3.739.644.646.350.764.691.998.599.898.592.229
Sm38 = p1 * p1 * p23 * p43
2 *
3 *
86893956354189878775643 *
2367958875411463048104007458352976869124861
Sm39 = p1 * p2 * p3 * p4 * p25 * p36
3 *
67 *
311 *
1039 *
6216157781332031799688469 *
305788363093026251381516836994235539
Sm40 = (p1)^2 * p1 * p4 * p5 * p6 * p10 * p20 * p26
2^2 *
5 *
3169 *
60757 *
579779 *4362289433 *
79501124416220680469 *
15944694111943672435829023
Sm41 = p1 * p3 * p6 * p8 * p56
3 *
487 *
493127 *
32002651 *53545135784961981058419604998638516483529257158438201753
Sm42 = p1 * p1 * p3 * p3 * p11 * p25 * p34
Page 12 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 13/32
1234567...43
7
17
449
231.058.353.953.907.153.927.797.941.629.430.896.528.
705.484.237.484.443.924.582.239.474.910.453
1234567...44
23
32
12.797.571.009.458.074.720.816.277
1.339.846.151.380.678.925.030.581.935.625.950.075.
102.697.197.563.351
1234567...45
32
5
7
41
727
1.291
2.634.831.682.519
379.655.178.169.650.473
10.181.639.342.830.457.495.311.038.751.840.866.580.
037
1234567...46
2
31
103
270.408.101
374.332.796.208.406.291
3.890.951.821.355.123.413.169.209
4.908.543.378.923.330.485.082.351.119
2 *
3 *
127 *
421 *
22555732187 *
4562371492227327125110177 *
3739644646350764691998599898592229
Sm43 = p1 * p2 * p3 * p72
7 *
17 *
449 *
231058353953907153927797941629430896528705484237484443924582239474910453
Sm44 = (p1)^3 * (p1)^2 * p26 * p52
2^3 *
3^2 *
12797571009458074720816277 *
1339846151380678925030581935625950075102697197563351
Sm45 = (p1)^2 * p1 * p1 * p2 * p3 * p4 * p13 * p18 * p41
3^2 *
5 *
7 *
41 *
727 *
1291 *
2634831682519 *
379655178169650473 *
10181639342830457495311038751840866580037
Sm46 = p1 * p2 * p3 * p9 * p18 * p25 * p28
2 *
31 *
103 *
270408101 *
374332796208406291 *
3890951821355123413169209 *
4908543378923330485082351119
Page 13 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 14/32
1234567...47
3
4.813
679.751
4.626.659.581.180.187.993.501
27.186.948.196.033.729.596.487.563.460.186.407.241.
534.572.026.740.723
1234567...48
22
3
179
1.493
1.894.439
15.771.940.624.188.426.710.323.588.657
1.288.413.105.003.100.659.990.273.192.963.354.903.
752.853.409
1234567...49
23
109
3.251.653
2.191.196.713
53.481.597.817.014.258.108.937
12.923.219.128.084.505.550.382.930.974.691.083.231.
834.648.599
1234567...50
2
3
52
13
211
20.479
160.189.818.494.829.241
46.218.039.785.302.111.919
19.789.860.528.346.995.527.543.912.534.464.764.790.
909.391
Sm47 = p1 * p4 * p6 * p22 * p53
3 *4813 *
679751 *
4626659581180187993501 *
27186948196033729596487563460186407241534572026740723
Sm48 = (p1)^2 * p1 * p3 * p4 * p7 * p29 * p462^2 *
3 *
179 *
1493 *
1894439 *
15771940624188426710323588657 *
1288413105003100659990273192963354903752853409
Sm49 = p2 * p3 * p7 * p10 * p23 * p47
23 *
109 *
3251653 *
2191196713 *
53481597817014258108937 *
12923219128084505550382930974691083231834648599
Sm50 = p1 * p1 * (p1)^2 * p2 * p3 * p5 * p18 * p20 * p44
2 *
3 *
5^2 *
13 *
211 *
20479 *
160189818494829241 *
46218039785302111919 *
19789860528346995527543912534464764790909391
Page 14 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 15/32
1234567...51
3
17.708.093.685.609.923.339
2.323.923.950.500.978.408.934.946.776.574.079.545.
611.397.611.995.364.705.071.565.292.612.305.003
1234567...52
27
43.090.793.230.759.613
2.238.311.464.092.386.636.761.884.511.894.978.048.
448.617.178.182.150.344.531.477.542.781.856.216.
843
1234567...53
33
73
127.534.541.853.151.177
1.045.271.879.581.348.729.278.017.817.925.065.799.
872.257.805.888.381.045.072.615.907.010.178.634.
849
1234567...54
2
36
79
389
3.167
13.309
69.526.661.707
8.786.705.495.566.261.913.717
107.006.417.566.370.797.549.761.092.803.112.128.112.
769.421.435.739
1234567...55
5
768.643.901
641.559.846.437.453
1.187.847.380.143.694.126.117
4.215.236.719.202.000.513.320.239.996.510.510.828.
557.825.033.460.062.191
Sm51 = p1 * p20 * p73
3 *
17708093685609923339 *
2323923950500978408934946776574079545611397611995364705071565292612305003
Sm52 = (p1)^7 * p17 * p76
2^7 *
43090793230759613 *
2238311464092386636761884511894978048448617178182150344531477542781856216843
Sm53 = (p1)^3 * (p1)^3 * p18 * p76
3^3 *
7^3 *
127534541853151177 *
1045271879581348729278017817925065799872257805888381045072615907010178634849
Sm54 = p1 * (p1)^6 * p2 * p3 * p4 * p5 * p11 * p22 * p51
2 *
3^6 *
79 *
389 *
3167 *
13309 *
69526661707 *
8786705495566261913717 *
107006417566370797549761092803112128112769421435739
Sm55 = p1 * p9 * p15 * p22 * p55
5 *
768643901 *
Page 15 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 16/32
1234567...56
22
3
4.324.751.743.617.631.024.407.823
23.788.800.764.365.032.854.813.369.830.458.732.886.
158.417.401.021.113.465.643.479.155.975.828.316.
681
1234567...57
3
17
36.769.067
2.205.251.248.721
2.128.126.623.795.388.466.914.401.931.224.151.279
14.028.351.843.196.901.173.601.082.244.449.305.344.
230.057.319
1234567...58
2
13
1.448.595.612.076.564.044.790.098.185.437
327.789.067.063.631.145.720.134.335.581.588.856.152.
921.479.945.230.066.396.717.484.857.630.796.759
1234567...59
3
340.038.104.073.949.513
324.621.819.487.091.567.830.636.828.971.096.713
3.728.107.520.554.143.574.058.126.525.447.653.708.
074.390.492.098.041.537
1234567...60
23
3
5
97
157
67.555.753.880.267.981.819.314.968.257.940.564.232.
852.139.165.917.171.861.439.543.181.780.049.107.
204.700.168.947.673.874.146.559.500.327
641559846437453 *
1187847380143694126117 *
4215236719202000513320239996510510828557825033460062191
Sm56 = (p1)^2 * p1 * p25 * p77
2^2 *
3 *
4324751743617631024407823 *
23788800764365032854813369830458732886158417401021113465643479155975828316681
Sm57 = p1 * p2 * p8 * p13 * p37 * p47
3 *
17 *
36769067 *
2205251248721 *
2128126623795388466914401931224151279 *
14028351843196901173601082244449305344230057319
Sm58 = p1 * p2 * p31 * p75
2 *
13 *
1448595612076564044790098185437 *
327789067063631145720134335581588856152921479945230066396717484857630796759
Sm59 = p1 * p18 * p36 * p55
3 *
340038104073949513 *
324621819487091567830636828971096713 *
3728107520554143574058126525447653708074390492098041537
Sm60 = (p1)^3 * p1 * p1 * p2 * p3 * p104
Page 16 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 17/32
1234567...61
10.386.763
35.280.457.769.357
33.689.963.756.771.087.787.406.890.988.794.422.071.
942.750.389.483.226.687.410.462.898.596.940.470.
571.223.420.915.460.371
1234567...62
2
32
1.709
329.167
1.830.733
9.703.956.232.921.821.226.401.223.348.541.281
6.862.941.251.271.421.600.892.952.202.464.376.235.
224.342.144.596.167.046.191.804.311
1234567...63
32
17.028.095.263
2.435.984.189.933.032.657.913.735.712.547.671.618.
367.909
330.698.276.590.517.405.413.770.500.371.046.766.676.563.523.569.978.590.938.716.221
1234567...64
22
7
17
19
197
522.673
1.072.389.445.090.071.30720.203.723.083.803.464.811.983.788.589
611.891.180.337.745.942.599.768.541.236.768.900.814.
521.123.060.392.220.304.537
2^3 *
3 *
5 *
97 *
157 *
67555753880267981819314968257940564232852139165917171861439543181780049107204700168947673874146559500327
Sm61 = p8 * p14 * p92
10386763 *
35280457769357 *
33689963756771087787406890988794422071942750389483226687410462898596940470571223420915460371
Sm62 = p1 * (p1)^2 * p4 * p6 * p7 * p34 * p64
2 *
3^2 *
1709 *
329167 *
1830733 *
9703956232921821226401223348541281 *
6862941251271421600892952202464376235224342144596167046191804311
Sm63 = (p1)^2 * p11 * p43 * p63
3^2 *
17028095263 *
2435984189933032657913735712547671618367909 *
330698276590517405413770500371046766676563523569978590938716221
Sm64 = (p1)^2 * p1 * p2 * p2 * p3 * p6 * p19 * p29 * p60
2^2 *
7 *
17 *
19 *
197 *
522673 *
1072389445090071307 *
20203723083803464811983788589 *
Page 17 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 18/32
1234567...65
3
5
31
83.719
8.018.741.962.917.674.781.000.851.595.476.715.337.
223.177
3.954.865.825.608.609.239.925.917.139.441.010.044.
747.553.878.722.812.487.568.124.023.324.127
1234567...66
2
3
7
20.143
971.077
319.873.117.219.722.504.963.051.951.872.747.251
927.600.480.728.565.729.398.211.282.118.577.179
506.464.674.142.683.362.314.480.915.373.647.544.917
1234567...67
397
183.783.139.772.372.071
169.207.186.381.096.030.569.641.287.629.182.352.063.847.752.831.832.860.300.985.727.686.482.291.228.
260.812.667.458.777.140.342.739.211.041
1234567...68
24
3
23
764.558.869
1.811.890.921
16.210.201.583.355.429.120.740.178.111.425.145.802.
012.035.286.59749.798.299.077.316.075.944.525.952.275.152.868.666.
920.234.906.076.151.289
1234567...69
611891180337745942599768541236768900814521123060392220304537
Sm65 = p1 * p1 * p2 * p5 * p43 * p70
3 *
5 *
31 *
83719 *
8018741962917674781000851595476715337223177 *
3954865825608609239925917139441010044747553878722812487568124023324127
Sm66 = p1 * p1 * p1 * p5 * p6 * p36 * p36 * p39
2 *
3 *
7 *
20143 *
971077 *
319873117219722504963051951872747251 *
927600480728565729398211282118577179 *
506464674142683362314480915373647544917
Sm67 = p3 * p18 * p105
397 *
183783139772372071 *
169207186381096030569641287629182352063847752831832860300985727686482291228260812667458777140342739211041
Sm68 = (p1)^4 * p1 * p2 * p9 * p10 * p50 * p56
2^4 *
3 *
23 *
764558869 *
1811890921 *
16210201583355429120740178111425145802012035286597 *
49798299077316075944525952275152868666920234906076151289
Page 18 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 19/32
3
13
23
8.684.576.204.660.284.317.187
281.259.608.597.535.749.175.083
15.490.495.288.652.004.091.050.327.089.107
3.637.485.176.043.309.178.386.946.614.318.767.365.
372.143.115.591
1234567...70
2
5
2.411.111
109.315.518.091.391.293.936.799
11.555.516.101.313.335.177.332.236.222.295.571.524.
323
405.346.669.169.620.786.437.208.619.979.711.016.226.
055.320.437.594.464.205.451
1234567...71
32
83
2.281
7.484.379.467.407.391.660.418.419.352.839
96.808.455.591.058.960.266.687.738.381.050.176.698.
103.277.406.505.724.847.082.994.829.643.349.780.
363.432.993.640.165.860.627
1234567...72
22
32
5.119
596.176.870.295.201.674.946.617.769
1.123.704.769.960.650.101.739.921.630.151.581.054.
522.510.738.566.183.226.239.911.321.871.780.637.
830.758.881.774.623.162.921.434.662.407
1234567...73 (by Philippe Strohl)
37.907
1.612.352.371.081.094.864.112.011.094.480.307.952.
600.705.089
201.992.666.185.187.831.800.817.490.810.938.117.880.
Sm69 = p1 * p2 * p2 * p22 * p24 * p32 * p49
3 *13 *
23 *
8684576204660284317187 *
281259608597535749175083 *
15490495288652004091050327089107 *
3637485176043309178386946614318767365372143115591
Sm70 = p1 * p1 * p7 * p24 * p41 * p60
2 *
5 *
2411111 *
109315518091391293936799 *
11555516101313335177332236222295571524323 *
405346669169620786437208619979711016226055320437594464205451
Sm71 = (p1)^2 * p2 * p4 * p31 * p95
3^2 *
83 *
2281 *
7484379467407391660418419352839 *
96808455591058960266687738381050176698103277406505724847082994829643349780363432993640165860627
Sm72 = (p1)^2 * (p1)^2 * p4 * p27 * p103
2^2 *
3^2 *
5119 *
596176870295201674946617769 *
1123704769960650101739921630151581054522510738566183226239911321871780637830758881774623162921434662407
Page 19 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 20/32
341.395.186.600.971.262.233.773.863.756.955.874.
363.353.778.851
1234567...74
2
3
71.788.313
21.565.573
99.014.155.049.267.797.799
1.634.187.291.640.507.800.518.363
1.981.231.397.449.722.872.290.863.561.307
2.377.534.541.508.613.492.655.260.491.688.014.802.
698.908.815.817
1234567...75 (by Sean A. Irvine)
3
52
193.283
38.824.496.309.870.038.690.197.243.565.592.769.246.
963.314.017
219.358.378.032.318.168.161.320.006.998.916.878.634.
145.966.511.629.131.235.131.312.083.699.783.021.
949.850.982.403
1234567...76
23
828.699.354.354.766.183
213.643.895.352.490.047.310.058.981
8.716.407.028.594.814.374.740.596.028.898.426.313.
034.395.366.012.872.513.707.917.231.855.753.694.
435.270.081.076.237.925.828.389
Sm73 = p5 * p46 * p87 ( Philippe Strohl )
37907 *
1612352371081094864112011094480307952600705089 *
201992666185187831800817490810938117880341395186600971262233773863756955874363353778851
Factor p46 Sm73 by GMP-ECM
Sm73 = 37907.p46.p87
None of these factors could have been found by P-1 or P+1 with B1<10^14 and I was lucky
enough to catch the p46 with a ECM B1 of 10^6.
The group order of the curve is very smooth (B1=620227 and B2=1473569 are enough).
325683354264679693500307906698027336176043019186246110832678756888/
805244789707561834881407263896785700945962383243895973215176272739 (132 digits)
Using B1=2000000, B2=5000000, polynomial x^6, sigma=2799427343
Step 1 took 181065ms
********** Factor found in step 1: 1612352371081094864112011094480307952600705089
Found probable prime factor of 46 digits: 1612352371081094864112011094480307952600705089
Probable prime cofactor 201992666185187831800817490810938117880341395186600971262233773863756955874363353778851
has 87 digits (both proven prime by S. Tomabechi APR-CL part of p_1 program)
Sm74 = p1 * p1 * p1 * p7 * p8 * p20 * p25 * p31 * p49
2 *
3 *
7 *
1788313 *
21565573 *
99014155049267797799 *
1634187291640507800518363 *
1981231397449722872290863561307 *2377534541508613492655260491688014802698908815817
Sm75 = p1 * (p1)^2 * p6 * p47 * p87 ( Sean A. Irvine )
3 *
5^2 *
193283 *38824496309870038690197243565592769246963314017 *
219358378032318168161320006998916878634145966511629131235131312083699783021949850982403
Sm76 = (p1)^3 * p18 * p27 * p97
Page 20 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 21/32
1234567...77
3
383.481.022.289.718.079.599.637
874.911.832.937.988.998.935.021
164.811.751.226.239.402.858.361.187.055.939.797.929
7.442.132.227.048.590.901.854.639.419.294.226.672.231.934.035.068.486.536.423
1234567...78 (by Sean A. Irvine)
2
3
31
185.897
205.155.431.830.422.787.082.756.234.197.593.935.249.
202.704.547.671.264.423
17.403.902.113.720.391.120.287.411.398.887.911.225.
298.966.708.915.583.006.414.519.403.038.472.992.
542.973.083
1234567...79
73
137
22.683.534.613.064.519.783
132.316.335.833.889.742.191.77335.488.612.864.124.533.038.957.177.977
11.589.330.059.060.921.218.833.486.882.285.427.414.
280.233.987.959.540.582.909.167.514.265.308.253
1234567...80
22
33
5
101
10.263.751
1.295.331.340.195.453.366.408.489
1.702.600.917.839.548.328.745.392.482.587.491.026.
230.318.172.323.434.581.398.602.992.701.169.952.
537.157.469.971.305.061.091.390.839.579.932.352.
102.383
2^3 *
828699354354766183 *
213643895352490047310058981 *
8716407028594814374740596028898426313034395366012872513707917231855753694435270081076237925828389
Sm77 = p1 * p24 * p24 * p39 * p58
3 *
383481022289718079599637 *
874911832937988998935021 *
164811751226239402858361187055939797929 *
7442132227048590901854639419294226672231934035068486536423
Sm78 = p1 * p1 * p2 * p6 * p57 * p83 ( Sean A. Irvine )
2 *
3 *
31 *
185897 *
205155431830422787082756234197593935249202704547671264423 *
17403902113720391120287411398887911225298966708915583006414519403038472992542973083
Sm79 = p2 * p3 * p20 * p24 * p29 * p74
73 *
137 *
22683534613064519783 *
132316335833889742191773 *
35488612864124533038957177977 *
11589330059060921218833486882285427414280233987959540582909167514265308253
Sm80 = (p1)^2 * (p1)^3 * p1 * p3 * p8 * p25 * p115
2^2 *
3^3 *
5 *
101 *
Page 21 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 22/32
1234567...81
33
509
152.873.624.211.113.444.108.313.548.197
58.762.581.888.644.185.603.361.112.342.786.137.599.
799.640.821.735.382.180.404.307.223.995.625.796.
855.706.598.141.292.123.658.134.092.320.545.833.186.103.011
1234567...82
2
29
4.703
10.091
12.295.349.967.251.726.424.104.854.676.730.107
334.523.571.229.968.373.890.203.385.137.399.026.475.051
1.090.461.105.551.993.653.223.776.199.179.348.475.
393.504.023.636.425.991.597.284.018.461.539
1234567...83 (by Sean A. Irvine)
3
53
503177.918.442.980.303.859
21.875.480.270.521.598.141.087.357.354.188.092.945.
840.550.359.281.483
3.966.169.790.267.211.790.412.249.283.896.602.109.
358.687.165.012.835.285.295.541.472.324.348.526.
743.126.307
1234567...84
25
3
128.600.821.980.325.136.890.793.456.450.022.106.587.
763.153.419.719.076.284.732.850.389.416.045.981.
702.547.359.113.015.678.672.244.328.809.985.375.
641.941.298.506.955.072.611.638.268.203.924.769.
581.335.2379
10263751 *
1295331340195453366408489 *
1702600917839548328745392482587491026230318172323434581398602992701169952537157469971305061091390839579932352102383
Sm81 = (p1)^3 * p3 * p30 * p119
3^3 *
509 *
152873624211113444108313548197 *
58762581888644185603361112342786137599799640821735382180404307223995625796855706598141292123658134092320545833186103
Sm82 = p1 * p2 * p4 * p5 * p35 * p42 * p70
2 *
29 *
4703 *
10091 *
12295349967251726424104854676730107 *
334523571229968373890203385137399026475051 *
1090461105551993653223776199179348475393504023636425991597284018461539
Sm83 = p1 * p2 * p3 * p18 * p53 * p82 ( Sean A. Irvine )
3 *
53 *
503 *
177918442980303859 *
21875480270521598141087357354188092945840550359281483 *
3966169790267211790412249283896602109358687165012835285295541472324348526743126307
by SNFS, 8 days
Sm84 = (p1)^5 * p1 * p157
2^5 *
3 *1286008219803251368907934564500221065877631534197190762847328503894160459817025473591130156786722443288099853756419412985069550726116382682
Page 22 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 23/32
1234567...85 (by Sean A. Irvine)
5
72
120.549.814.855.596.987.772.827.562.271.063.563.633.
851.059
2.112.809.210.944.968.177.871.685.727.287.164.545.
437.750.155.430.310.661
197.843.626.412.162.026.434.764.405.036.310.959.588.
059.884.460.495.810.550.047
1234567...86 (by Sean A. Irvine)
2
323
1.056.149
718.252.229.986.396.496.762.902.999.331.863.301.257
10.828.687.641.092.318.839.822.035.841.363.590.407.
263.202.742.239.027.773
1.089.075.252.400.674.157.091.531.724.111.232.381.
528.208.779.232.955.680.665.273
1234567...87 (by Sean A. Irvine)
3
7
231.330.259
4.275.444.601
101.784.611.215.757.903.569.658.774.280.830.604.745.
279.416.597.473
58.398.250.025.786.270.255.235.847.423.735.930.777.
973.447.337.337.804.788.906.368.149.837.276.410.
666.257.137.526.766.841.721
1234567...88 (by Sean A. Irvine)
22
Sm85 = p1 * (p1)^2 * p45 * p55 * p60 ( Sean A. Irvine )
5 *
7^2 *
120549814855596987772827562271063563633851059 *
2112809210944968177871685727287164545437750155430310661 *
197843626412162026434764405036310959588059884460495810550047
Sm85 C158 =
120549814855596987772827562271063563633851059 (p45) *
Using B1=11000000, sigma=1708124291
2112809210944968177871685727287164545437750155430310661 (p55) *
197843626412162026434764405036310959588059884460495810550047 (p60)
by GNFS, 1 day
Sm86 = p1 * p1 * p2 * p7 * p39 * p56 * p61 ( Sean A. Irvine )
2 *
3 *
23 *
1056149 *
718252229986396496762902999331863301257 *
10828687641092318839822035841363590407263202742239027773 *
1089075252400674157091531724111232381528208779232955680665273
Sm86 C154 =
718252229986396496762902999331863301257 (p39) * C116
Using B1=11000000, sigma=3414478964
10828687641092318839822035841363590407263202742239027773 (p56) *
1089075252400674157091531724111232381528208779232955680665273 (p61)
by GNFS, 2 days.
Sm87 = p1 * p1 * p9 * p10 * p51 * p95 ( Sean A. Irvine )
3 *
7 *
231330259 *
4275444601 *
101784611215757903569658774280830604745279416597473 *
58398250025786270255235847423735930777973447337337804788906368149837276410666257137526766841721
Sm87 C145 = (p51) * (p95)
by SNFS, 14 days.
Page 23 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 24/32
12.414.068.351.873
462.668.377.429.470.430.246.269.302.055.630.668.010.
673
144.494.999.796.935.291.164.027.251.780.366.969.508.
458.166.480.331
3.718.931.833.006.826.909.360.514.481.439.595.803.
175.244.655.637.881.136.348.103
1234567...89 (by Sean A. Irvine)
32
13
31
97
163.060.459
789.841.356.493.369.879496.118.159.817.126.721.484.175.235.476.073
26.459.905.787.227.421.825.352.754.831.024.262.009.
257
2.075.552.579.046.417.801.880.667.285.191.357.553.
672.027.185.826.871.770.761.977.511
1234567...90 (by Sean A. Irvine)
2
32
5
1.987
179.827
2.166.457
5.469.640.487.155.071.172.064.105.436.159.054.827.
205.011.884.517.193.846.381.587.779.057
323.974.513.721.871.489.318.385.733.207.245.357.406.
204.798.917.206.286.895.918.649.972.193.592.038.
458.818.136.011
1234567...91 (by Sean A. Irvine)
37
Sm88 = (p1)^2 * p14 * p42 * p51 * p61 ( Sean A. Irvine )
2^2 *
12414068351873 *462668377429470430246269302055630668010673 *
144494999796935291164027251780366969508458166480331 *
3718931833006826909360514481439595803175244655637881136348103
Sm88 C153=
462668377429470430246269302055630668010673 (p42)
B1=11000000, sigma=1512552247
144494999796935291164027251780366969508458166480331 (p51) *
3718931833006826909360514481439595803175244655637881136348103 (p61)
by GNFS, 8 hours
Sm89 = (p1)^2 * p2 * p2 * p2 * p9 * p18 * p33 * p41 * p64 ( Sean A. Irvine )
3^2 *
13 *
31 *
97 *
163060459 *
789841356493369879 *
496118159817126721484175235476073 *
26459905787227421825352754831024262009257 *
2075552579046417801880667285191357553672027185826871770761977511
Sm90 = p1 * (p1)^2 * p1 * p4 * p6 * p7 * p67 * p87 ( Sean A. Irvine )
2 *
3^2 *5 *
1987 *
179827 *
2166457 *
5469640487155071172064105436159054827205011884517193846381587779057 *
323974513721871489318385733207245357406204798917206286895918649972193592038458818136011
Sm90 C154=
(p67) * (p87)
by SNFS, 32 days
Submitted on Monday October 24, 2005 22:51
Page 24 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 25/32
607
5.713.601.747.802.353
100.397.446.615.566.314.002.487
3.581.874.457.050.057.021.838.729.610.409.482.762.
969.149.632.972.915.379
267.535.593.139.950.330.755.907.265.689.770.024.664.
090.795.106.497.661.308.268.157.342.396.003.221
1234567...92
23
3
75.503
46.731.404.628.893.905.607.210.235.741.707 ( 'p32' by Sean A. Irvine)
17.357.685.121.487.530.272.314.084.020.479.969.142.
526.171.001.787.819.150.223.751.641
839.921.864.959.969.600.234.341.350.615.454.280.584.
339.900.783.049.158.479.018.433.912.354.703
1234567...93
3
73
1.051
3.298.142.203
19.544.056.951.015.647.623.992.763.251 ( 'p29' by Sean A. Irvine)
4.886.013.639.051.371.332.965.225.321.191.263.200.
785.903.705.285.317
1.703.057.751.798.522.700.187.996.077.196.637.285.
517.155.003.415.445.664.199.429.017.748.369.723.
643.706.497
1234567...94 (by Greg Childers)
2
12.871.181
98.250.285.823
1.825.097.233.762.709.447.432.521.941.926.649.289.
213.154.260.264.910.537.140.594.516.431.173.070.
300.371
Sm91 = p2 * p3 * p16 * p24 * p55 * p75 ( Sean A. Irvine )
37 *
607 *5713601747802353 *
100397446615566314002487 *
3581874457050057021838729610409482762969149632972915379 *
267535593139950330755907265689770024664090795106497661308268157342396003221
Sm91 C129=
(p55) * (p75)
by GNFS, 4 days
Submitted on Monday October 24, 2005 22:51
Sm92 = (p1)^3 * p1 * p5 * p32 * p65 * p72 ( Sean A. Irvine )
2^3 *
3 *
75503 *
46731404628893905607210235741707 *
17357685121487530272314084020479969142526171001787819150223751641 *
839921864959969600234341350615454280584339900783049158479018433912354703
Sm92 C137=
(p65) * (p72)
by GNFS, 9 days
Submitted on Sunday January 22, 2006 21:28
Sm93 = p1 * p2 * p4 * p10 * p29 * p52 * p82 ( Sean A. Irvine )
3 *
73 *
1051 *
3298142203 *
19544056951015647623992763251 *
4886013639051371332965225321191263200785903705285317 *1703057751798522700187996077196637285517155003415445664199429017748369723643706497
Sm93 C133=
(p52) * (p82)
by GNFS, 5 days
Submitted on Monday February 20, 2006 23:01
Page 25 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 26/32
2.674.525.573.684.858.697.560.701.870.658.348.933.
916.102.325.593.721.165.422.426.453.989.766.526.
938.215.889
1234567...95 (by Sean A. Irvine)
3
5
7
401
244.987.542.265.129.586.458.446.183.157.595.351
119.684.333.324.585.760.380.296.925.278.736.677.052.
991.667.067.598.465.535.119.086.641.122.308.977.
254.652.550.763.964.697.554.302.296.677.991.161.
440.001.789.403.458.655.109.609.795.769
1234567...96
22
3
23
60.331
7.414.218.343.605.898.007.054.904.008.539.678.229.
463.872.328.651.811.494.111.562.828.507.144.051.
357.405.695.052.612.835.346.584.059.319.708.614.758.837.877.621.899.193.657.692.066.488.505.067.
022.654.601.125.869.790.297.498.349.041
1234567...97
13
949.667.608.470.093.318.578.167.063.015.547.864.032.
712.517.561.002.409.487.257.972.106.456.954.941.
803.426.651.911.500.396.348.881.197.366.045.850.
894.335.742.820.591.305.439.790.288.275.136.759.985.244.833.729.682.214.530.699.379.184.227.669
1234567...98 (by Philippe Strohl)
2
32
23
37
Sm94 = p1 * p8 * p11 * p79 * p82 ( Greg Childers )
2 *
12871181 *
98250285823 *
1825097233762709447432521941926649289213154260264910537140594516431173070300371 *
2674525573684858697560701870658348933916102325593721165422426453989766526938215889
Summary for Sm94(c160) = p79 * p82
The factorization was completed using SNFS. GGNFS was used for the sieving
and msieve for the post-processing.
Submitted on Sun, 9 Dec 2007 11:27
Sm95 = p1 * p1 * p1 * p3 * p36 * p141 ( Sean A. Irvine )
3 *
5 *
7 *
401 *
244987542265129586458446183157595351 *
11968433332458576038029692527873667705299166706759846553511908664112230897725465255076396469755430229667799116144000
Sm96 = (p1)^2 * p1 * p2 * p5 * p175
2^2 *
3 *
23 *
60331 *
7414218343605898007054904008539678229463872328651811494111562828507144051357405695052612835346584059319708614758837877621899193657692066488
Sm97 = p2 * p183
13 *
9496676084700933185781670630155478640327125175610024094872579721064569549418034266519115003963488811973660458508943357428205913054397902882
Page 26 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 27/32
199
1.495.444.452.918.817
381.502.754.125.464.943.168.932.369.122.248.696.781
3.588.472.635.471.667.861.938.967.869.443.938.442.
910.813.342.994.227.048.889
197.825.482.406.769.698.151.783.117.995.020.967.519.
766.027.202.915.861.687.264.259.155.363
1234567...99 (by Greg Childers)
32
31.601
786.576.340.181
37.726.668.883.887.938.032.416.757.819.314.355.053.
940.153.680.075.342.644.295.667.759
14.627.910.783.072.606.795.565.990.651.314.126.145.
674.770.336.615.677.946.549.896.262.532.933.945.
988.541.999.815.567.058.347.827.465.728.809
1234567...100 (by Sean A. Irvine)
22
52
73
8.171
1.065.829
2.824.782.749
20.317.177.407.273.276.661
970.447.246.795.177.523.033.247.400.823
7.420.578.382.899.399.028.284.464.392.651.452.937.
744.039.836.185.355.778.662.961.413.780.805.734.
369.643.748.805.299.589.898.776.112.804.950.234.
221.784.569
1234567...101
3
8.377
799.917.088.062.980.754.649
6.141.298.867.893.783.540.378.996.188.437.492.127.
764.003.798.590.081.909.642.744.192.552.148.805.
391.571.094.462.286.943.973.793.319.310.495.244.
Sm98 = p1 * (p1)^2 * p2 * p2 * p3 * p16 * p39 * p58 * p69 ( Philippe Strohl )
2 *
3^2 *23 *
37 *
199 *
1495444452918817 *
381502754125464943168932369122248696781 *
3588472635471667861938967869443938442910813342994227048889 *
197825482406769698151783117995020967519766027202915861687264259155363
Sm99 = (p1)^2 * p5 * p12 * p65 * p107 ( Greg Childers )
3^2 *
31601 *
786576340181 *
37726668883887938032416757819314355053940153680075342644295667759 *
14627910783072606795565990651314126145674770336615677946549896262532933945988541999815567058347827465728809
Summary for Sm99(c177) = p65 * p107
I finally got around to factoring Sm99 by SNFS. As for Sm94, I used
the GGNFS lattice siever and msieve for the postprocessing.
Submitted on Tue, 20 May 2008 4:35
Sm100 = (p1)^2 * (p1)^2 * (p1)^3 * p4 * p7 * p10 * p20 * p30 * p118 ( Sean A. Irvine )
2^2 *
5^2 *
7^3 *
8171 *
1065829 *
2824782749 *
20317177407273276661 *
970447246795177523033247400823 *
74205783828993990282844643926514529377440398361853557786629614137808057343696437488052995898987761128049502342217845
Page 27 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 28/32
779.697.771.718.230.720.369.645.297.842.348.701.
438.176.773.337.763.517.045.479 = c169
1234567...102 2
3
19
89
3.607
15.887
32.993
2.865.523.753
2.245.981.950.884.772.863.770.930.273.540.385.579.
914.865.629.636.627.917.458.256.811.732.689.892.
492.870.743.326.877.749.976.350.147.897.124.023.
992.523.914.020.180.640.624.011.740.696.205.659.
507.665.744.332.920.411.510.673.767
1234567...103
131
1.231
1.713.675.826.579.469
446.742.179.244.293.705.107.743.562.882.710.303.107.
653.012.176.267.248.963.278.610.465.019.097.212.
371.922.408.861.122.959.220.266.036.734.062.945.
987.327.999.759.759.889.653.172.480.328.343.669.
298.241.710.606.083.565.775.177.315.278.367 = c180
1234567...104
26
3
59
773
19.601.852.982.312.892.289
719.258.686.675.513.979.262.824.247.665.486.572.432.830.986.721.319.631.951.392.684.768.588.734.879.
818.050.477.878.475.895.676.175.151.494.878.249.
756.161.105.289.114.143.497.898.340.693.385.704.
920.480.426.209.979.668.623.253.319.269 = c177
Sm101 = p1 * p4 * p21 * c169
3 *
8377 *
799917088062980754649 *
6141298867893783540378996188437492127764003798590081909642744192552148805391571094462286943973793319310495244779697771718230720369645297842
Sm102 = p1 * p1 * p2 * p2 * p4 * p5 * p5 * p10 * p172
2 *3 *
19 *
89 *
3607 *
15887 *
32993 *
2865523753 *
2245981950884772863770930273540385579914865629636627917458256811732689892492870743326877749976350147897124023992523914020180640624011740696
Sm103 = p3 * p4 * p16 * c180
131 *
1231 *
1713675826579469 *
4467421792442937051077435628827103031076530121762672489632786104650190972123719224088611229592202660367340629459873279997597598896531724803
Sm104 = (p1)^6 * p1 * p2 * p3 * p20 * c177
2^6 *
3 *
59 *
773 *
19601852982312892289 *
7192586866755139792628242476654865724328309867213196319513926847685887348798180504778784758956761751514948782497561611052891141434978983406
Page 28 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 29/32
1234567...105
3
5
193
6.942.508.281.251
6.142.568.425.081.438.832.759.347.097.237.645.657.
628.988.841.290.897.653.306.131.105.424.225.283.
715.083.312.222.699.695.626.373.378.408.508.801.
576.676.945.473.675.560.832.710.564.855.374.168.
538.899.701.991.629.924.122.849.992.620.409.192.891.550.349 = c190
1234567...106 (by Sean A. Irvine)
2
11
127
827
95.383.501.607.400.293.616.004.374.931
54.259.599.094.002.572.583.355.411.045.946.413
375.159.085.605.310.877.928.459.072.269.605.386.
653.376.782.374.874.196.433.925.741.599.663
27.518.056.325.201.854.933.261.643.718.251.313.
697.576.510.084.474.601.978.478.694.683.051.
383
1234567...107
33
536.288.185.369
8.526.150.295.974.562.563.911.703.097.396.807.303.
361.305.853.752.080.385.827.103.422.281.006.173.
895.434.732.314.352.853.475.423.512.542.010.066.
856.002.013.066.381.244.223.149.688.686.332.747.
287.256.098.942.256.562.363.655.334.309.484.941.
298.623.600.483.738.889
1234567...108 (by Sean A. Irvine)
22
33
128.451.681.010.379.681
132.761.751.746.390.611.923.240.080.737.166.083
67.031.425.578.179.280.405.553.486.489.006.742.336.
953.759.049.830.840.809.351.016.348.413.007.664.
845.819.742.768.984.976.575.205.426.833.399.525.
010.462.614.317.613.333.284.615.639.359.796.130.
220.299.502.987.337
Sm105 = p1 * p1 * p3 * p13 * c190
3 *
5 *
193 *
6942508281251 *
6142568425081438832759347097237645657628988841290897653306131105424225283715083312222699695626373378408508801576676945473675560832710564855
Sm106 = p1 * p2 * p3 * p3 * p29 * p35 * p69 * p71 ( Sean A. Irvine )
2 *
11 *
127 *
827 *
95383501607400293616004374931 *
54259599094002572583355411045946413 *
375159085605310877928459072269605386653376782374874196433925741599663 *
27518056325201854933261643718251313697576510084474601978478694683051383
by GNFS, 7 days
Finally did another of these numbers, sorry but it is not the most wanted Sm101.
Submitted on Wed, 29 Apr 2009 17:50
Sm107 = (p1)^3 * p12 * p199
3^3 *
536288185369 *
8526150295974562563911703097396807303361305853752080385827103422281006173895434732314352853475423512542010066856002013066381244223149688686
Page 29 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 30/32
Please doublecheck the correctness of the above results before using them for continuing the search!
In the Queue
Sm116 (COMPLETE) by Sean A. Irvine
2^2
3^2
2239
9787002048140152171263515060558503699 (p37)
1564979683672455156550590564550896170739590132221492657705868045238053081655725205100502721369811982/50507087284878063256342705928229557508508670247743582143974583381133763456377474127925121483818271 (p198)
Sm117 (PARTIAL) by Philippe Strohl
3^2
31883
333699561211(p12 by MF)
28437086452217952631(p20 by MF)
4533906451347845613823069537391478156136844054596308469800656338667810383531584928935260534747032327/
7527359676572226592530190953518990632828989750268624468732498680573884588879299814792334945634701707/
955171 (c206)
Philippe Strohl found a new factor of Sm117 (but the cofactor is still composite) :
( Friday 22/08/2008 14:59 )
Input number is above c206
Using B1=50000000, B2=288591693406, polynomial Dickson(12), sigma=759744520
dF=65536, k=6, d=690690, d2=17, i0=56
Expected number of curves to find a factor of n digits:
20 25 30 35 40 45 50 55 60 65
2 5 14 51 223 1139 6555 42004 296146 2292504
Step 1 took 1567405msStep 2 took 365869ms
********** Factor found in step 2: 29899433706805424728763564400367447
Found probable prime factor of 35 digits: 29899433706805424728763564400367447
Composite cofactor
1516385392381488800257172455421115218103131389426237636403907504872104848630821256747576627427151045/
955402969807173970504574397911322632329216437824800943241454211577975893 has 172 digits
Sm121 (COMPLETE) by Sean A. Irvine
278240783 (p9)
105299178204417486675841093021769 (p33)
4213754301973277818574830150933029703205115128282586723382785882706969263182976786615125991432774212/
6655712800813928005415583544197992453104126217919256625510887081121101381586161564163756343745220847/
88731721938623 (p214)
Sm148 (COMPLETE) by Sean A. Irvine
2^2
197
1192717377
273131
623321
3417425341307 (p13)
4614988413949 (p13)
8817212782626223819399721069204897 (p34)
3193000701568524782467188898304641220775712837053116231323237434768208956576768718690200934704769644/
9776432217795787176033049303491281548912080640497966801122571925082634457098946350721137505551941519/
011986808243341521869976182605502561225915860092642869 (p254)
Sm152 (COMPLETE) by Sean A. Irvine
2^4
3^2
131
10613
29354379044409991753 (p20)
Sm108 = (p1)^2 * (p1)^3 * p18 * p36 * p161 ( Sean A. Irvine )
2^2 *
3^3 *
128451681010379681 *
132761751746390611923240080737166083 *
6703142557817928040555348648900674233695375904983084080935101634841300766484581974276898497657520542683339952501046261431761333328461563935
Page 30 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 31/32
2587833772662908004979 (p22)
4103096315830350734534473515557 (p31)
12805089500421274253268517941967 (p32)
17815076027044127272632744936161 (p32)
8672648427724666836335878649605123533671234498113722493001839423884394310675246313883662523667972796/
2252207354099527091658621300178181661297993537192234834905032751669182605720711181867690701061985005/
06817 (p205)
Sm165 (COMPLETE) by Sean A. Irvine
3
5
7
1331
247007767 (p9)
490242053931613 (p15)
13183356310254866666237435750357 (p32)
1827567681941731356121062274519777291863760972834724140109364732283149431121763995026237820114303681/
5496592791501562871236022199555066011298451642231401735631929300761746199878863922607830081439132951/
8357914807533066417404260975659341568447505060701439696780455579283391217843114550193398184405873168/
7369979309788684753888188553 (p328)
Sm184 (COMPLETE) by Sean A. Irvine
2^4
7
59
191
1093
1223
22521973429 (p11)
15219125459582087 (p17)
158906425126963139 (p18)
2513521443592870099 (p19)
677008100402429325901609057 (p27)
7894977574571781556444786202593614139721506720817604366015528775760676373160442530345019483307700978/
5930092303778152032404551795675090170923053660293230843498614359663939844365627366165832785305317365/
4182408958317242742517820581180854453052925226658686768857580470091786086406610221754789129568203967/
384451608838167466879488313009807568569387 (p342)
Sm187 (COMPLETE) by Sean A. Irvine
349
506442073 (p9)
1080829169904060835770214147747 (p31)
6462532135259365632021314942658431728094733620599149141734327082367671298692320282350900597278296365/
5379540884023312710555856177308446767405172970938977772676796780228431702242816509113421339444592236/
2621714833233212554723714564174418111669498936207951085298551799080803363445759267522417246541605647/
9089775584237803310812087978174533031535543826808011950270774768093377786126458352214138913849333920/
84296657173 (p411)
Sm195 (COMPLETE) by Sean A. Irvine
3
5397
21728563
300856949 (p9)
554551531 (p9)
8174619091 (p10)
165897663095213559529993681 (p27)
4216891792160044902686705799521388925390732888122432608757782570720072408665875338945809590873483499/
4620499297336767795766591884079938389340512146288914907796490815205714454677249295016313996731519073/
1294500128685930803732434591580562083296784964092846142348549703545534554252170080984646226664593569/
3244989301840859149448482745301257117142121991254187915811979621816086743861383233522991211424294391/
495728519167 (p412)
Page 31 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...
8/12/2010http://users.skynet.be/worldofnumbers/factorlist.htm
8/9/2019 All the prime factors of the Smarandache Concatenated Numbers upto the first not factor
http://slidepdf.com/reader/full/all-the-prime-factors-of-the-smarandache-concatenated-numbers-upto-the-first 32/32
[ TOP OF PAGE]
(All rights reserved) - Last modified : June 8, 2009.
Patrick De Geest - Belgium - Short Bio - Some Pictures E-mail address : [email protected]
Page 32 of 32All the prime factors of the Smarandache Concatenated Numbers upto the first not fact ...