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Normal Smarandache Concatenated Numbers
Prime factors from 1 upto n (n=2,3,...,200)
rood Reversed Smarandache Concatenated Numbers
rood Repunits Factorization rood comments


Itinerary


sm1sm2sm3sm4sm5sm6sm7sm8sm9sm10sm11sm12sm13sm14sm15sm16sm17sm18sm19sm20
sm21sm22sm23sm24sm25sm26sm27sm28sm29sm30sm31sm32sm33sm34sm35sm36sm37sm38sm39sm40
sm41sm42sm43sm44sm45sm46sm47sm48sm49sm50sm51sm52sm53sm54sm55sm56sm57sm58sm59sm60
sm61sm62sm63sm64sm65sm66sm67sm68sm69sm70sm71sm72sm73sm74sm75sm76sm77sm78sm79sm80
sm81sm82sm83sm84sm85sm86sm87sm88sm89sm90sm91sm92sm93sm94sm95sm96sm97sm98sm99sm100
sm101sm102sm103sm104sm105sm106sm107sm108sm109sm110sm111sm112sm113sm114sm115sm116sm117sm118sm119sm120
sm121sm122sm123sm124sm125sm126sm127sm128sm129sm130sm131sm132sm133sm134sm135sm136sm137sm138sm139sm140
sm141sm142sm143sm144sm145sm146sm147sm148sm149sm150sm151sm152sm153sm154sm155sm156sm157sm158sm159sm160
sm161sm162sm163sm164sm165sm166sm167sm168sm169sm170sm171sm172sm173sm174sm175sm176sm177sm178sm179sm180
sm181sm182sm183sm184sm185sm186sm187sm188sm189sm190sm191sm192sm193sm194sm195sm196sm197sm198sm199sm200

Legend
sm_Nif N less than 104 - complete factorization is given on this page
sm_Nif N is above 104 - refer for the complete factorization to M. Fleuren page
sm_Nsmallest Sm with unknown factors
sm_NSm with unknown factors (ref. M. Fleuren's page)
sm_NSm with new complete factorization : see list at end of page
sm_NSm with a new factor but still incomplete : consult Messages section
sm_NSm is prime (smallest one > Sm38712 ref. E. Weisstein) !


Prefatory Notes & Sources

In the table below you'll find all the prime factors of the concatenation
of numbers from 1 upto n.
These numbers are called  Smarandache Concatenated Numbers. 

The first one with an unknown prime factor is when n = 104.
If there is a breaktrough in partially or completely factorising Sm104, please let me know,
so that I can update the list.

For the factorizations I also followed the sources from
Micha Fleuren, Smarandache factors
Hans Havermann, Factorization of Smarandache Concatenated Numbers, Sm-n (n < 84)

Other subject related sources on the web :

Smarandache Numbers by Dr. M. L. Perez
Smarandache factors by Micha Fleuren
Primes by Listing by Carlos Rivera
Consecutive Number Sequences by Eric W. Weisstein
Smarandache Sequences by Eric W. Weisstein
List of factors of the Reversed Smarandache Concatenated Numbers by Patrick De Geest

Book sources :

"Some Notions and Questions in Number Theory", by C.Dumitrescu and V.Seleacu, Glendale, AZ:Erhus University Press, 1994.
(communicated to me by Marin Petrescu (email) from Bucharest)

"CRC Concise Encyclopedia of Mathematics", by Eric W. Weisstein, CRC Press, Boca Raton, Florida, 1998.
(communicated to me by M.L. Perez (email))

OEIS entries :

A007908 - Concatenation of the numbers from 1 to n.
A046460 - Number of prime factors of concatenation of numbers from 1 up to n, with multiplicity.
A046461 - Numbers n such that concatenation of numbers from 1 to n is a semiprime.
A046462 - Concatenation of numbers from 1 to a(n) has exactly 3 prime factors, with multiplicity.
A046463 - Concatenation of numbers from 1 to a(n) has exactly 4 prime factors, with multiplicity.
A046464 - Concatenation of numbers from 1 to a(n) has exactly 5 prime factors, with multiplicity.
A046465 - Concatenation of numbers from 1 to a(n) has exactly 6 prime factors, with multiplicity.
A046466 - Concatenation of numbers from 1 to a(n) has exactly 7 prime factors, with multiplicity.
A046467 - Concatenation of numbers from 1 to a(n) has exactly 8 prime factors, with multiplicity.
A046468 - Concatenation of numbers from 1 to a(n) has exactly 9 prime factors, with multiplicity.
A048342 - Numbers n such that the concatenation of the numbers 1, 2, ..., n is a product of distinct primes.
A050675 - Numbers n such that concatenation of numbers from 1 to n is a powerful(1) number.
A050676 - Let b(n) = number of prime factors (with multiplicity) of concatenation of numbers from 1 to n; sequence gives smallest number m with b(m) = n.


Some Factorization Websites

Factorization using the Elliptic Curve Method
GGNFS - A Number Field Sieve implementation


Messages

[ April 25, 2015 ]
Stephen Tucker's (email) search for prime Smarandache numbers !

Dear Patrick,
I have found your list of Smaradache factors and noticed that no prime Smarandache number
has been found yet. Well, I decided to try looking for one.
Using Dario Alpern's ECM Factorizing applet, (and software I wrote myself to generate the numbers),
I have discovered that there are no prime Smarandache numbers less than Sm2659.
When I tried using Dario's applet to factorize Sm2659 (which, by the way, has no factors less than
or equal to 39989) the applet's attempt to start the Prime Check routine stalled.
I tried using it to check Sm2713, but the same thing happened again.
Dario's website does stipulate a maximum length of input number of 10000 digits. However, Sm2659
is "only" 9529 digits long, so perhaps his stated limit of 10000 is rounded up.
After a brief search on the web, I haven't discovered anything about prime Smaradanche numbers.
I wonder if it could be that a Smaradanche number cannot be prime.

Hope this is of interest.

Yours,
Stephen Tucker (UK).

Answer
Eric Weisstein [http://mathworld.wolfram.com/ConsecutiveNumberSequences.html]
wrote he extended the search up to 38712 terms which is
quite ahead of your Sm2659. He did find a probable prime for
the reversed case though (Rsm37765).

Note Primeform with the program PFGW64.EXE has a built-in command
Sm(x) and Smr(x) to search for (probable) primes directly.
I did a run up to Sm(10000) and found indeed none. For larger
values one needs a faster computer than I have at the moment.
So there is still opportunity to detect the first PRP Sm !


[ 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 used
the 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 sieved
to 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 Smarandache
concatenated number. It was a 126 digits composite.
I have broken it with ggnfs (a wonderful program : I have done it with
a common laptop computer with "only" 512 megs of ram and a 1.4 GHz
celeron 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 =
(p51) * (p95)
by SNFS, 14 days
Sm88 C153 =
(p42) * (p51) * (p61)
by GNFS, 8 hours

Sm90 looks like it will have to be SNFS since ECM has failed
to 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* 2.3^2.23.37.199
p16: 1495444452918817(MF)
c165: 270825497607069872452415496119443135107702791840293286471110488510
4768274391266695197120574357173627794391936143016235446328574795690351940341420
23605896434694145167

Line=16/32 Curves=47/1000 B1=1000000 factors=0
C165 Using B1=1000000, B2=839549780, polynomial Dickson(6), sigma=594918519
Step 1 took 28422ms
Step 2 took 18926ms
********** Factor found in step 2: 381502754125464943168932369122248696781
Found probable prime factor of 39 digits:
381502754125464943168932369122248696781

Composite cofactor
709891330215674922888729762564179876071621230485156504316706784486089628388958197669654729613064737223993515869067455247541707
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 earlier
I 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=1708124291
2112809210944968177871685727287164545437750155430310661 (p55) *
197843626412162026434764405036310959588059884460495810550047 (p60)
by GNFS, 1 day

Sm86 C154 =
718252229986396496762902999331863301257 (p39) * C116
Using B1=11000000, sigma=3414478964
C116 by GNFS nearly done

Sm87 C145 Sieving by SNFS started 2005-08-29.

Others with B1=1e6 (I have now completed 1000 curves with B1=1e6
on all Sm(n), n <= 200)

Sm114 8678622406220213516465050301044327
Sm159 45941358846148651407783221723920871719
Sm171 40202471819457246557501649563881337
Sm193 5167315927941164272437909427556797

Sean.

It seems that a fierce competition is going on. But to avoid
duplicate work and loose valuable cpu time I advice strongly
to 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 by
ecm) and Sm86 (with a p39 and a ggnfs on the remaining c115)... This
represents quite a large amount of cpu work. Sm84 factorization is still
known 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 seen
previously reported. All these were found with ECM B1=1e6.

Sm89 496118159817126721484175235476073
Sm89 26459905787227421825352754831024262009257.P64
Sm92 46731404628893905607210235741707
Sm93 19544056951015647623992763251

Sm95 244987542265129586458446183157595351.P141
Sm100 970447246795177523033247400823.P118
Sm106 95383501607400293616004374931
Sm106 54259599094002572583355411045946413

Sm108 132761751746390611923240080737166083.P161
Sm109 9943216978062352390003139833531
Sm114 2042059881000388200555074336219

Sm116 9787002048140152171263515060558503699.P198
Sm121 105299178204417486675841093021769.P214
Sm123 12347002211187670552593982429
Sm123 2829927788416784955921382453753
Sm125 295999706346724665505289
Sm137 144065103514544138702103468451

Sm148 8817212782626223819399721069204897.P254
Sm152 4103096315830350734534473515557
Sm152 12805089500421274253268517941967
Sm152 17815076027044127272632744936161.P205
Sm154 32063206397901252963254536935569
Sm159 11855111297257593607972759339201
Sm160 64603936118676024484144135734907
Sm162 22260247937572504750086047
Sm164 1039418554780603268384723777072953

Sm165 13183356310254866666237435750357.P328
Sm176 1011379313630785579015894871
Sm183 553245689211853052761209813199

Sm184 677008100402429325901609057.P342
Sm187 1080829169904060835770214147747.P411
Sm193 419908232491384495189
Sm195 165897663095213559529993681.P412
Sm198 14158849264684185910199571953

Further, after studying Backstrom's work on Rsm76 I am now able to generate
SNFS polynomials for all the remaining Sm numbers below 100. It would have
been much faster to do Sm78 by SNFS, but I had already started it before
working out how to apply SNFS to the number. It should be possible to complete
all 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.


[ 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 complete
with the discovery of a p46 by GMP-ECM...

Sm73 = 37907.p46.p87
p46: 1612352371081094864112011094480307952600705089
p87: 201992666185187831800817490810938117880341395186600971262233773863756955874363353778851
...
Sm74 factorization is known and the next composite to challenge is Sm75
with no factors expected below 35 digits...

Sm74 = 2.3.7.1788313.21565573.p20.p25.p31.p49
p20: 99014155049267797799
p25: 1634187291640507800518363(PZ)
p31: 1981231397449722872290863561307
p49: 2377534541508613492655260491688014802698908815817

Sm75* 3.5^2.193283.c133
c133:
851647853845481367839983983361331811035304896846801931077529055832/
3936344974946612980172082837107906069172212808249295700548030242851



The List of Sm Factors

 1
1

Sm1 = p1 = unity 
1
12 22 3
Sm2 = (p1)^2 * p1
2^2 *
3
123 3 41
Sm3 = p1 * p2 = semiprime 
3 *
41
1234 2 617
Sm4 = p1 * p3 = semiprime 
2 *
617
12345 3 5 823
Sm5 = p1 * p1 * p3
3 *
5 *
823
123456 26 3 643
Sm6 = (p1)^6 * p1 * p3
2^6 *
3 *
643
1234567 127 9.721
Sm7 = p3 * p4 = semiprime 
127 *
9721
12345678 2 32 47 14.593
Sm8 = p1 * (p1)^2 * p2 * p5
2 *
3^2 *
47 *
14593
123456789 32 3.607 3.803
Sm9 = (p1)^2 * p4 * p4
3^2 *
3607 *
3803
12345678910 2 5 1.234.567.891
Sm10 = p1 * p1 * p10
2 *
5 *
1234567891
1234567891011 3 7 13 67 107 630.803
Sm11 = p1 * p1 * p2 * p2 * p3 * p6
3 *
7 *
13 *
67 *
107 *
630803
1234567...12 23 3 2.437 2.110.805.449
Sm12 = (p1)^3 * p1 * p4 * p10
2^3 *
3 *
2437 *
2110805449
1234567...13 113 125.693 869.211.457
Sm13 = p3 * p6 * p9
113 *
125693 *
869211457
1234567...14 2 3 205.761.315.168.520.219
Sm14 = p1 * p1 * p18
2 *
3 *
205761315168520219
1234567...15 3 5 8.230.452.606.740.808.761
Sm15 = p1 * p1 * p19
3 *
5 *
8230452606740808761
1234567...16 22 2.507.191.691 1.231.026.625.769
Sm16 = (p1)^2 * p10 * p13
2^2 *
2507191691 *
1231026625769
1234567...17 32 47 4.993 584.538.396.786.764.503
Sm17 = (p1)^2 * p2 * p4 * p18
3^2 *
47 *
4993 *
584538396786764503
1234567...18 2 32 97 88.241 801.309.546.900.123.763
Sm18 = p1 * (p1)^2 * p2 * p5 * p18
2 *
3^2 *
97 *
88241 *
801309546900123763
1234567...19 13 43 79 281 1.193 833.929.457.045.867.563
Sm19 = p2 * p2 * p2 * p3 * p4 * p18
13 *
43 *
79 *
281 *
1193 *
833929457045867563
1234567...20 25 3 5 323.339 3.347.983 2.375.923.237.887.317
Sm20 = (p1)^5 * p1 * p1 * p6 * p7 * p16
2^5 *
3 *
5 *
323339 *
3347983 *
2375923237887317
1234567...21 3 17 37 43 103 131 140.453 802.851.238.177.109.689
Sm21 = p1 * p2 * p2 * p2 * p3 * p3 * p6 * p18
3 *
17 *
37 *
43 *
103 *
131 *
140453 *
802851238177109689
1234567...22 2 7 1.427 3.169 85.829 2.271.991.367.799.686.681.549
Sm22 = p1 * p1 * p4 * p4 * p5 * p22
2 *
7 *
1427 *
3169 *
85829 *
2271991367799686681549
1234567...23 3 41 769 13.052.194.181.136.110.820.214.375.991.629
Sm23 = p1 * p2 * p3 * p32
3 *
41 *
769 *
13052194181136110820214375991629
1234567...24 22 3 7 978.770.977.394.515.241 1.501.601.205.715.706.321
Sm24 = (p1)^2 * p1 * p1 * p18 * p19
2^2 *
3 *
7 *
978770977394515241 *
1501601205715706321
1234567...25 52 15.461 31.309.647.077 1.020.138.683.879.280.489.689.401
Sm25 = (p1)^2 * p5 * p11 * p25
5^2 *
15461 *
31309647077 *
1020138683879280489689401
1234567...26 2 34 21.347 2.345.807 982.658.598.563 154.870.313.069.150.249
Sm26 = p1 * (p1)^4 * p5 * p7 * p12 * p18
2 *
3^4 *
21347 *
2345807 *
982658598563 *
154870313069150249
1234567...27 33 192 4.547 68.891 40.434.918.154.163.992.944.412.000.742.833
Sm27 = (p1)^3 * (p2)^2 * p4 * p5 * p32
3^3 *
19^2 *
4547 *
68891 *
40434918154163992944412000742833
1234567...28 23 47 409 416.603.295.903.037 192.699.737.522.238.137.890.605.091
Sm28 = (p1)^3 * p2 * p3 * p15 * p27
2^3 *
47 *
409 *
416603295903037 *
192699737522238137890605091
1234567...29 3 859 24.526.282.862.310.130.729 19.532.994.432.886.141.889.218.213
Sm29 = p1 * p3 * p20 * p26
3 *
859 *
24526282862310130729 *
19532994432886141889218213
1234567...30 2 3 5 13 49.269.439 370.677.592.383.442.753 17.333.107.067.824.345.178.861
Sm30 = p1 * p1 * p1 * p2 * p8 * p18 * p23
2 *
3 *
5 *
13 *
49269439 *
370677592383442753 *
17333107067824345178861
1234567...31 29 2.597.152.967 163.915.283.880.121.143.989.433.769.727.058.554.332. 117
Sm31 = p2 * p10 * p42
29 *
2597152967 *
163915283880121143989433769727058554332117
1234567...32 22 3 7 45.068.391.478.912.519.182.079 326.109.637.274.901.966.196.516.045.637
Sm32 = (p1)^2 * p1 * p1 * p23 * p30
2^2 *
3 *
7 *
45068391478912519182079 *
326109637274901966196516045637
1234567...33 3 23 269 7.547 116.620.853.190.351.161 7.557.237.004.029.029.700.530.634.132.859
Sm33 = p1 * p2 * p3 * p4 * p18 * p31
3 *
23 *
269 *
7547 *
116620853190351161 *
7557237004029029700530634132859
1234567...34 2 6.172.839.455.055.606.570.758.085.909.601.061.116. 212.631.364.146.515.661.667
Sm34 = p1 * p58 = semiprime
2 *
6172839455055606570758085909601061116212631364146515661667
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
Sm35 = (p1)^2 * p1 * p3 * p3 * p8 * p10 * p37
3^2 *
5 *
139 *
151 *
64279903 *
4462548227 *
4556722495899317991381926119681186927
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
Sm36 = (p1)^4 * (p1)^2 * p3 * p3 * p56
2^4 *
3^2 *
103 *
211 *
39448709943503776711542648338171477043440283875433388943
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
Sm37 = p2 * p5 * p7 * p52
71 *
12379 *
4616929 *
3042410911077206144807069396988766146557218727107817
1234567...38 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
Sm38 = p1 * p1 * p23 * p43
2 *
3 *
86893956354189878775643 *
2367958875411463048104007458352976869124861
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
Sm39 = p1 * p2 * p3 * p4 * p25 * p36
3 *
67 *
311 *
1039 *
6216157781332031799688469 *
305788363093026251381516836994235539
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
Sm40 = (p1)^2 * p1 * p4 * p5 * p6 * p10 * p20 * p26
2^2 *
5 *
3169 *
60757 *
579779 *
4362289433 *
79501124416220680469 *
15944694111943672435829023
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
Sm41 = p1 * p3 * p6 * p8 * p56
3 *
487 *
493127 *
32002651 *
53545135784961981058419604998638516483529257158438201753
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
Sm42 = p1 * p1 * p3 * p3 * p11 * p25 * p34
2 *
3 *
127 *
421 *
22555732187 *
4562371492227327125110177 *
3739644646350764691998599898592229
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
Sm43 = p1 * p2 * p3 * p72
7 *
17 *
449 *
231058353953907153927797941629430896528705484237484443924582239474910453
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
Sm44 = (p1)^3 * (p1)^2 * p26 * p52
2^3 *
3^2 *
12797571009458074720816277 *
1339846151380678925030581935625950075102697197563351
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
Sm45 = (p1)^2 * p1 * p1 * p2 * p3 * p4 * p13 * p18 * p41
3^2 *
5 *
7 *
41 *
727 *
1291 *
2634831682519 *
379655178169650473 *
10181639342830457495311038751840866580037
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
Sm46 = p1 * p2 * p3 * p9 * p18 * p25 * p28
2 *
31 *
103 *
270408101 *
374332796208406291 *
3890951821355123413169209 *
4908543378923330485082351119
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
Sm47 = p1 * p4 * p6 * p22 * p53
3 *
4813 *
679751 *
4626659581180187993501 *
27186948196033729596487563460186407241534572026740723
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
Sm48 = (p1)^2 * p1 * p3 * p4 * p7 * p29 * p46
2^2 *
3 *
179 *
1493 *
1894439 *
15771940624188426710323588657 *
1288413105003100659990273192963354903752853409
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
Sm49 = p2 * p3 * p7 * p10 * p23 * p47
23 *
109 *
3251653 *
2191196713 *
53481597817014258108937 *
12923219128084505550382930974691083231834648599
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
Sm50 = p1 * p1 * (p1)^2 * p2 * p3 * p5 * p18 * p20 * p44
2 *
3 *
5^2 *
13 *
211 *
20479 *
160189818494829241 *
46218039785302111919 *
19789860528346995527543912534464764790909391
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
Sm51 = p1 * p20 * p73
3 *
17708093685609923339 *
2323923950500978408934946776574079545611397611995364705071565292612305003
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
Sm52 = (p1)^7 * p17 * p76
2^7 *
43090793230759613 *
2238311464092386636761884511894978048448617178182150344531477542781856216843
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
Sm53 = (p1)^3 * (p1)^3 * p18 * p76
3^3 *
7^3 *
127534541853151177 *
1045271879581348729278017817925065799872257805888381045072615907010178634849
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
Sm54 = p1 * (p1)^6 * p2 * p3 * p4 * p5 * p11 * p22 * p51
2 *
3^6 *
79 *
389 *
3167 *
13309 *
69526661707 *
8786705495566261913717 *
107006417566370797549761092803112128112769421435739
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
Sm55 = p1 * p9 * p15 * p22 * p55
5 *
768643901 *
641559846437453 *
1187847380143694126117 *
4215236719202000513320239996510510828557825033460062191
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
Sm56 = (p1)^2 * p1 * p25 * p77
2^2 *
3 *
4324751743617631024407823 *
23788800764365032854813369830458732886158417401021113465643479155975828316681
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
Sm57 = p1 * p2 * p8 * p13 * p37 * p47
3 *
17 *
36769067 *
2205251248721 *
2128126623795388466914401931224151279 *
14028351843196901173601082244449305344230057319
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
Sm58 = p1 * p2 * p31 * p75
2 *
13 *
1448595612076564044790098185437 *
327789067063631145720134335581588856152921479945230066396717484857630796759
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
Sm59 = p1 * p18 * p36 * p55
3 *
340038104073949513 *
324621819487091567830636828971096713 *
3728107520554143574058126525447653708074390492098041537
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
Sm60 = (p1)^3 * p1 * p1 * p2 * p3 * p104
2^3 *
3 *
5 *
97 *
157 *
67555753880267981819314968257940564232852139165917171861439543181780049107204700168947673874146559500327
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
Sm61 = p8 * p14 * p92
10386763 *
35280457769357 *
33689963756771087787406890988794422071942750389483226687410462898596940470571223420915460371
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
Sm62 = p1 * (p1)^2 * p4 * p6 * p7 * p34 * p64
2 *
3^2 *
1709 *
329167 *
1830733 *
9703956232921821226401223348541281 *
6862941251271421600892952202464376235224342144596167046191804311
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
Sm63 = (p1)^2 * p11 * p43 * p63
3^2 *
17028095263 *
2435984189933032657913735712547671618367909 *
330698276590517405413770500371046766676563523569978590938716221
1234567...64 22 7 17 19 197 522.673 1.072.389.445.090.071.307 20.203.723.083.803.464.811.983.788.589 611.891.180.337.745.942.599.768.541.236.768.900.814. 521.123.060.392.220.304.537
Sm64 = (p1)^2 * p1 * p2 * p2 * p3 * p6 * p19 * p29 * p60
2^2 *
7 *
17 *
19 *
197 *
522673 *
1072389445090071307 *
20203723083803464811983788589 *
611891180337745942599768541236768900814521123060392220304537
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
Sm65 = p1 * p1 * p2 * p5 * p43 * p70
3 *
5 *
31 *
83719 *
8018741962917674781000851595476715337223177 *
3954865825608609239925917139441010044747553878722812487568124023324127
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
Sm66 = p1 * p1 * p1 * p5 * p6 * p36 * p36 * p39
2 *
3 *
7 *
20143 *
971077 *
319873117219722504963051951872747251 *
927600480728565729398211282118577179 *
506464674142683362314480915373647544917
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
Sm67 = p3 * p18 * p105
397 *
183783139772372071 *
169207186381096030569641287629182352063847752831832860300985727686482291228260812667458777140342739211041
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. 012.035.286.597 49.798.299.077.316.075.944.525.952.275.152.868.666. 920.234.906.076.151.289
Sm68 = (p1)^4 * p1 * p2 * p9 * p10 * p50 * p56
2^4 *
3 *
23 *
764558869 *
1811890921 *
16210201583355429120740178111425145802012035286597 *
49798299077316075944525952275152868666920234906076151289
1234567...69 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
Sm69 = p1 * p2 * p2 * p22 * p24 * p32 * p49
3 *
13 *
23 *
8684576204660284317187 *
281259608597535749175083 *
15490495288652004091050327089107 *
3637485176043309178386946614318767365372143115591
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
Sm70 = p1 * p1 * p7 * p24 * p41 * p60
2 *
5 *
2411111 *
109315518091391293936799 *
11555516101313335177332236222295571524323 *
405346669169620786437208619979711016226055320437594464205451
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
Sm71 = (p1)^2 * p2 * p4 * p31 * p95
3^2 *
83 *
2281 *
7484379467407391660418419352839 *
96808455591058960266687738381050176698103277406505724847082994829643349780363432993640165860627
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
Sm72 = (p1)^2 * (p1)^2 * p4 * p27 * p103
2^2 *
3^2 *
5119 *
596176870295201674946617769 *
1123704769960650101739921630151581054522510738566183226239911321871780637830758881774623162921434662407
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. 341.395.186.600.971.262.233.773.863.756.955.874. 363.353.778.851
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)
1234567...74 2 3 7 1.788.313 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
Sm74 = p1 * p1 * p1 * p7 * p8 * p20 * p25 * p31 * p49
2 *
3 *
7 *
1788313 *
21565573 *
99014155049267797799 *
1634187291640507800518363 *
1981231397449722872290863561307 *
2377534541508613492655260491688014802698908815817
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
Sm75 = p1 * (p1)^2 * p6 * p47 * p87    ( Sean A. Irvine )
3 *
5^2 *
193283 *
38824496309870038690197243565592769246963314017 *
219358378032318168161320006998916878634145966511629131235131312083699783021949850982403
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
Sm76 = (p1)^3 * p18 * p27 * p97
2^3 *
828699354354766183 *
213643895352490047310058981 *
8716407028594814374740596028898426313034395366012872513707917231855753694435270081076237925828389
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
Sm77 = p1 * p24 * p24 * p39 * p58
3 *
383481022289718079599637 *
874911832937988998935021 *
164811751226239402858361187055939797929 *
7442132227048590901854639419294226672231934035068486536423
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
Sm78 = p1 * p1 * p2 * p6 * p57 * p83    ( Sean A. Irvine )
2 *
3 *
31 *
185897 *
205155431830422787082756234197593935249202704547671264423 *
17403902113720391120287411398887911225298966708915583006414519403038472992542973083
1234567...79 73 137 22.683.534.613.064.519.783 132.316.335.833.889.742.191.773 35.488.612.864.124.533.038.957.177.977 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
Sm79 = p2 * p3 * p20 * p24 * p29 * p74
73 *
137 *
22683534613064519783 *
132316335833889742191773 *
35488612864124533038957177977 *
11589330059060921218833486882285427414280233987959540582909167514265308253
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
Sm80 = (p1)^2 * (p1)^3 * p1 * p3 * p8 * p25 * p115
2^2 *
3^3 *
5 *
101 *
10263751 *
1295331340195453366408489 *
1702600917839548328745392482587491026230318172323434581398602992701169952537157469971305061091390839579932352102383
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
Sm81 = (p1)^3 * p3 * p30 * p119
3^3 *
509 *
152873624211113444108313548197 *
58762581888644185603361112342786137599799640821735382180404307223995625796855706598141292123658134092320545833186103011
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
Sm82 = p1 * p2 * p4 * p5 * p35 * p42 * p70
2 *
29 *
4703 *
10091 *
12295349967251726424104854676730107 *
334523571229968373890203385137399026475051 *
1090461105551993653223776199179348475393504023636425991597284018461539
1234567...83 (by Sean A. Irvine) 3 53 503 177.918.442.980.303.859 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
Sm83 = p1 * p2 * p3 * p18 * p53 * p82    ( Sean A. Irvine )
3 *
53 *
503 *
177918442980303859 *
21875480270521598141087357354188092945840550359281483 *
3966169790267211790412249283896602109358687165012835285295541472324348526743126307
by SNFS, 8 days
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
Sm84 = (p1)^5 * p1 * p157
2^5 *
3 *
1286008219803251368907934564500221065877631534197190762847328503894160459817025473591130156786722443288099853756419412985069550726116382682039247695813352379
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
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
1234567...86 (by Sean A. Irvine) 2 3 23 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
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.
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
Sm87 = p1 * p1 * p9 * p10 * p51 * p95    ( Sean A. Irvine )
3 *
7 *
231330259 *
4275444601 *
101784611215757903569658774280830604745279416597473 *
58398250025786270255235847423735930777973447337337804788906368149837276410666257137526766841721
Sm87 C145 = (p51) * (p95)
by SNFS, 14 days.
1234567...88 (by Sean A. Irvine) 22 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
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
1234567...89 (by Sean A. Irvine) 32 13 31 97 163.060.459 789.841.356.493.369.879 496.118.159.817.126.721.484.175.235.476.073 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
Sm89 = (p1)^2 * p2 * p2 * p2 * p9 * p18 * p33 * p41 * p64    ( Sean A. Irvine )
3^2 *
13 *
31 *
97 *
163060459 *
789841356493369879 *
496118159817126721484175235476073 *
26459905787227421825352754831024262009257 *
2075552579046417801880667285191357553672027185826871770761977511
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
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
1234567...91 (by Sean A. Irvine) 37 607 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
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
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
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
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
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
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 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
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
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
Sm95 = p1 * p1 * p1 * p3 * p36 * p141    ( Sean A. Irvine )
3 *
5 *
7 *
401 *
244987542265129586458446183157595351 *
119684333324585760380296925278736677052991667067598465535119086641122308977254652550763964697554302296677991161440001789403458655109609795769
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
Sm96 = (p1)^2 * p1 * p2 * p5 * p175
2^2 *
3 *
23 *
60331 *
7414218343605898007054904008539678229463872328651811494111562828507144051357405695052612835346584059319708614758837877621899193657692066488505067022654601125869790297498349041
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
Sm97 = p2 * p183 = semiprime
13 *
949667608470093318578167063015547864032712517561002409487257972106456954941803426651911500396348881197366045850894335742820591305439790288275136759985244833729682214530699379184227669
1234567...98 (by Philippe Strohl) 2 32 23 37 199 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
Sm98 = p1 * (p1)^2 * p2 * p2 * p3 * p16 * p39 * p58 * p69    ( Philippe Strohl )
2 *
3^2 *
23 *
37 *
199 *
1495444452918817 *
381502754125464943168932369122248696781 *
3588472635471667861938967869443938442910813342994227048889 *
197825482406769698151783117995020967519766027202915861687264259155363
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
Sm99 = (p1)^2 * p5 * p12 * p65 * p107 [ Length = 189 ]    ( 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
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
Sm100 = (p1)^2 * (p1)^2 * (p1)^3 * p4 * p7 * p10 * p20 * p30 * p118 [ Length = 192 ]    ( Sean A. Irvine )
2^2 *
5^2 *
7^3 *
8171 *
1065829 *
2824782749 *
20317177407273276661 *
970447246795177523033247400823 *
7420578382899399028284464392651452937744039836185355778662961413780805734369643748805299589898776112804950234221784569
1234567...101 (by Bob Backstrom) 3 8.377 799.917.088.062.980.754.649 1.399.463.086.740.105.394.672.913.130.945.493.026. 937.913.499.238.148.790.743.003 4.388.325.012.701.307.167.526.588.635.576.876.644. 759.452.668.196.597.056.747.408.345.988.387.366. 211.263.062.577.487.913.664.612.635.611.915.493
Sm101 = p1 * p4 * p21 * p61 * p109 [ Length = 195 ]    ( Bob Backstrom )
3 *
8377 *
799917088062980754649 *
1399463086740105394672913130945493026937913499238148790743003 *
4388325012701307167526588635576876644759452668196597056747408345988387366211263062577487913664612635611915493
Summary for Sm101(c169) = p61 * p109

Hello Patrick,

Here's another wanted number for your tables:

Sm(101) = 3 * 8377 * 799917088062980754649 * C169
Tue Jun 5 01:18:11 2012 prp61 factor: 1399463086740105394672913130945493026937913499238148790743003
Tue Jun 5 01:18:11 2012 prp109 factor: 4388325012701307167526588635576876644759452668196597056747408345988387366211263062577487913664612635611915493
Tue Jun 5 01:18:11 2012 elapsed time 04:27:08
(Just the elapsed time for ONE sqrt - it only took one, luckily).

The whole number took many weeks on various machines. The relations were slow coming because the Snfs coefficients were pretty dreadful
as you can see from the Msieve log below.
Mon Jun 4 20:51:03 2012 Msieve v. 1.44
Mon Jun 4 20:51:03 2012 random seeds: 4714fab8 a98d82fd
Mon Jun 4 20:51:03 2012 factoring
6141298867893783540378996188437492127764003798590081909642744192552148805391571094462286943973793319/
310495244779697771718230720369645297842348701438176773337763517045479 (169 digits)
Mon Jun 4 20:51:04 2012 no P-1/P+1/ECM available, skipping
Mon Jun 4 20:51:04 2012 commencing number field sieve (169-digit input)
Mon Jun 4 20:51:04 2012 R0: -10000000000000000000000000000000000000
Mon Jun 4 20:51:04 2012 R1: 1
Mon Jun 4 20:51:04 2012 A0: -8919910099
Mon Jun 4 20:51:04 2012 A1: 0
Mon Jun 4 20:51:04 2012 A2: 0
Mon Jun 4 20:51:04 2012 A3: 0
Mon Jun 4 20:51:04 2012 A4: 0
Mon Jun 4 20:51:04 2012 A5: 12099999899800
Mon Jun 4 20:51:04 2012 skew 1.00, size 3.077673e-16, alpha -0.346420, combined = 6.775396e-13

Kind regards,
--Bob.

Submitted on Mon, 4 June 2012 18:44
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
Sm103 = p1 * p1 * p2 * p2 * p4 * p5 * p5 * p10 * p172 [ Length = 198 ]
2 *
3 *
19 *
89 *
3607 *
15887 *
32993 *
2865523753 *
2245981950884772863770930273540385579914865629636627917458256811732689892492870743326877749976350147897124023992523914020180640624011740696205659507665744332920411510673767
1234567...103 (by Sean A. Irvine) 131 1.231 1.713.675.826.579.469 16.908.963.624.339.537.484.508.436.321.314.327.604. 030.763.349.996.047.014.668.841.426.185.197 26.420.435.289.199.660.352.290.245.657.167.852.985. 476.641.946.070.651.819.895.933.156.168.339.498. 719.086.012.326.404.560.442.282.402.403.559.611
Sm103 = p3 * p4 * p16 * p71 * p110 [ Length = 201 ]    ( Sean A. Irvine )
131 *
1231 *
1713675826579469 *
16908963624339537484508436321314327604030763349996047014668841426185197 *
26420435289199660352290245657167852985476641946070651819895933156168339498719086012326404560442282402403559611
Summary for Sm103(c180) = p71 * p110

The factorization of the C180 after removal of the small factors was completed by SNFS using yafu.
The entire factorization took 6 months of otherwise idle time on a single 12-core machine.

Regards,
Sean A. Irvine

Submitted on Mon, 15 February 2016 10:46
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
Sm104 = (p1)^6 * p1 * p2 * p3 * p20 * c177 [ Length = 204 ]
2^6 *
3 *
59 *
773 *
19601852982312892289 *
719258686675513979262824247665486572432830986721319631951392684768588734879818050477878475895676175151494878249756161105289114143497898340693385704920480426209979668623253319269
Please doublecheck the correctness of the above results before using them for continuing the search!







In the Queue
 1234567...105  (by Sean A. Irvine)
3
5
193
6.942.508.281.251
90.853.974.148.830.729.568.788.807.471.204.169.448.
   373.857
67.609.243.102.773.972.838.875.424.854.217.967.300.
   371.972.209.133.190.536.893.586.620.791.162.850.
   744.838.052.281.507.779.485.845.273.498.264.830.
   080.938.632.761.526.794.830.712.920.440.816.557

Sm105 = p1 * p1 * p3 * p13 * p44 * p146 [ length = 207 ]    ( Sean A. Irvine )
3 *
5 *
193 *
6942508281251 *
90853974148830729568788807471204169448373857 *
67609243102773972838875424854217967300371972209133190536893586620791162850744838052281507779485845273498264830080938632761526794830712920440816557
Hi Patrick,
It has been a long time between drinks, but I finally factored another of these numbers
Sm105(c190)
Using B1=11000000, B2=35133391030, polynomial Dickson(12), sigma=982310399
Step 1 took 63945ms
Step 2 took 30405ms
********** Factor found in step 2: 90853974148830729568788807471204169448373857
Found probable prime factor of 44 digits: 90853974148830729568788807471204169448373857
Probable prime cofactor 67609243102773972838... ...94830712920440816557 has 146 digits
Sean.

Submitted on Wed, 22 Dec 2010 19:43
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
Sm106 = p1 * p2 * p3 * p3 * p29 * p35 * p69 * p71 [ Length = 210 ]    ( 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
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
Sm107 = (p1)^3 * p12 * p199 [ Length = 213 ]
3^3 *
536288185369 *
8526150295974562563911703097396807303361305853752080385827103422281006173895434732314352853475423512542010066856002013066381244223149688686332747287256098942256562363655334309484941298623600483738889
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
Sm108 = (p1)^2 * (p1)^3 * p18 * p36 * p161 [ Length = 216 ]    ( Sean A. Irvine )
2^2 *
3^3 *
128451681010379681 *
132761751746390611923240080737166083 *
67031425578179280405553486489006742336953759049830840809351016348413007664845819742768984976575205426833399525010462614317613333284615639359796130220299502987337
1234567...109 (by ?) 7 1.559 78.176.687 73.024.355.266.099.724.939 1.981.644.878.781.747.880.741.435.124.851.080.494. 790.628.315.732.046.577.374.765.000.215.165.197.716. 618.676.290.959.628.177.059.771.965.724.800.964.648. 196.375.253.775.228.530.930.793.786.414.419.275.506. 860.611.603.294.305.701.123.211.714.745.801 = c187
Sm109 = p1 * p4 * p8 * p20 * c187 [ Length = 219 ]    ( ? )
7 *
1559 *
78176687 *
73024355266099724939 *
1981644878781747880741435124851080494790628315732046577374765000215165197716618676290959628177059771965724800964648196375253775228530930793786414419275506860611603294305701123211714745801
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 1567405ms Step 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 11927 17377 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) 2587833772662908004979 (p22) 4103096315830350734534473515557 (p31) 12805089500421274253268517941967 (p32) 17815076027044127272632744936161 (p32) 8672648427724666836335878649605123533671234498113722493001839423884394310675246313883662523667972796/ 2252207354099527091658621300178181661297993537192234834905032751669182605720711181867690701061985005/ 06817 (p205) Sm165 (COMPLETE) by Sean A. Irvine 3 5 7 13 31 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 5 397 21728563 300856949 (p9) 554551531 (p9) 8174619091 (p10) 165897663095213559529993681 (p27) 4216891792160044902686705799521388925390732888122432608757782570720072408665875338945809590873483499/ 4620499297336767795766591884079938389340512146288914907796490815205714454677249295016313996731519073/ 1294500128685930803732434591580562083296784964092846142348549703545534554252170080984646226664593569/ 3244989301840859149448482745301257117142121991254187915811979621816086743861383233522991211424294391/ 495728519167 (p412)
Sm110 2.3.5.4517 p20: 18443752916913621413(AM) p197: 493962905754780705799621937897055143771136965793911814385622095572 1104630814091495547537529237766969832421058041142883772410973358977043070523990 1861854012027457023299672370583841892589110518827197 Sm111 3.293.431.230273.209071.241423723 p10: 3182306131(MF) p12: 171974155987(MF) p13: 1532064083461(MF) p17: 59183601887848987(MF) p19: 8526805649394145853(AM) p23: 27151072184008709784271(AM) p109: 244048003428987182237086269388683512609917095222912916711908327706 2899175394632300484951689048576681026896223 Sm112 2^3.16619.449797.894009023 p17: 74225338554790133(MF) p23: 10021106769497255963093(MF) p169: 310451505082372390807690913759034364782526954531565202979092678318 8211767084523827184031625338265911008653113512314794480936566758254656863951748 098953803988065923879729 Sm113* 3.11.13.5653.1016453.16784357 p18: 116507891014281007(AM) p37: 6844495453726387858061775603297883751(AM) c157: 374188046120925456299396712130641236411843308036898519351500335817 3776240679161689963186489546428779155940057835457190245380104884209783412515388 402615451617 Sm114* 2.3.7.178333 c227: 164829131991851695604186333848983459146989897288099488080943724559 4843623772253318827945673750596523887462352541330032480256663626703344138103432 7540543043523026436616702744076417445503890007284539931463704352198670078837278 349 Sm115* 5.17.19.41.36606.71518987 p18: 283858194594979819(AM) c202: 250883248398493472146591443544788529836402273741166377381726821922 1612944135548938768639596565770054053019051398713012801704173134789057321318643 580529670344967998709782174681880994644472022676758389491 Sm116* 2^2.3^2.2239 c235: 153164593694000460789987740300755821453343044120552718516876999123 9137066925145718137418677131722427617209889839415706994276671997756596911763284 4374731009426933984403874611594207433499565666728835058198613605541549812553981 32997284429 Sm117* 3^2.31883 p12: 333699561211(MF) p20: 28437086452217952631(MF) c206: 453390645134784561382306953739147815613684405459630846980065633866 7810383531584928935260534747032327752735967657222659253019095351899063282898975 0268624468732498680573884588879299814792334945634701707955171 Sm118* 2.83 p11: 33352084523(MF) p20: 20481677004050305811(MF) c214: 108872549668899506591755529039821174203503347054583911435720779733 0428976086992860487685496852884467613284688099978929533430468893051793566503012 287207422945984682148992628368919209392043127040432150185424056721141 Sm119* 3.59.101.139.2801 c239: 177375008101978180732720165303274790060342245926324757109737009411 6487799575622366019728505603894343116478279007942404241094231156311629167836263 0605443361422347301284854160351578178778682625109356092258844542785861818452104 859596299949073

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Patrick De Geest - Belgium flag - Short Bio - Some Pictures
E-mail address : pdg@worldofnumbers.com