[QUOTE=swishzzz;346865]How do I find the group order for this factorization? factordb's group order calculator clearly fails for this one...[/QUOTE]
sigma 1:<something> is not a Brent-Suyama curve, and Factordb up to now only supports Brent-Suyama curves (sigma 0:<something>).

Tried some Magma code, based on the GMP-ECM manual: (won't work, and I have no idea why)
[code]FindGroupOrder := function (p, s)
K := GF(p);
A := K ! (4*s^2-2);
x := 2;
b := x^3 + A*x^2 + x;
E := EllipticCurve([0,b*A,0,b^2,0]);
return FactoredOrder(E);
end function;

p := 26759964491830480636236398774973830719679139755537527;
s := 3576746370;
FindGroupOrder(p,s);[/code]

I've just got factorization of 102^103 + 1 = 103^2 * prp74 * prp130 with SNFS:
[code]prp74 factor: 16577923085747542727498881886756397313868752518022676502052070512564532587
prp130 factor: 4371325251720559422253332573045929417056159075217221344318685656228760863793509418168865967108430523971884258595890393501349300723
[/code]

A [URL="http://www.factordb.com/index.php?id=1100000000636053336"]P58[/URL] I've found by ECM last week (which broke my previous personal record):
[CODE]
GMP-ECM 6.4.4 [configured with MPIR 2.6.0] [ECM]
Input number is (3*10^204+7)/(2111*12379*80141*360193*36529453*83762087274812203759) (160 digits)
Using B1=260000000, B2=3178559884516, polynomial Dickson(30), sigma=2078429522
Step 1 took 1093707ms
Step 2 took 363420ms
********** Factor found in step 2: [URL="http://www.factordb.com/index.php?id=1100000000636053336"]4269986142493572515510539041322472993083083125849142037361[/URL]
Found probable prime factor of 58 digits: 4269986142493572515510539041322472993083083125849142037361
Probable prime cofactor ((3*10^204+7)/(2111*12379*80141*360193*36529453*83762087274812203759))/4269986142493572515510539041322472993083083125849142037361 has 102 digits
[/CODE]Details [URL="http://homepage2.nifty.com/m_kamada/math/c/30007.htm#N204_C160"]here[/URL].


A [URL="http://www.factordb.com/index.php?id=1100000000636107512"]P53[/URL] found by ECM in step 1 (details [URL="http://homepage2.nifty.com/m_kamada/math/c/28887.htm#N238_C206"]here[/URL]):
[CODE]
GMP-ECM 6.4.4 [configured with MPIR 2.6.0] [ECM]
Input number is (26*10^238-17)/(9*3*31*17914895525348997871953180891109) (206 digits)
Using B1=110000000, B2=776278396540, polynomial Dickson(30), sigma=4255384633
Step 1 took 648620ms
********** Factor found in step 1: [URL="http://www.factordb.com/index.php?id=1100000000636107512"]77799771931273889262077139536983524215512156277756839[/URL]
Found probable prime factor of 53 digits: 77799771931273889262077139536983524215512156277756839
Probable prime cofactor ((26*10^238-17)/(9*3*31*17914895525348997871953180891109))/77799771931273889262077139536983524215512156277756839 has 153 digits
[/CODE]

[CODE]Msieve v. 1.52 (SVN 886M)
Mon Nov 11 01:03:57 2013
random seeds: 187198dc e395e0be
factoring 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999992999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 (253 digits)
...initial square root is modulo 4585129
sqrtTime: 7150
p117 factor: 211857628677636264166676902114066256198845941934180788645905936104472665184004014226679863567582409810768749673482061
p137 factor: 47201510100993602326827859852217555163129145401663287219390072454758015933440888931227171966024510895342063068118093723212737563878094459
[/CODE]

[CODE]GMP-ECM 6.4.4 [configured with MPIR 2.6.0] [ECM]
Input number is (10^248+3)/(19*223*126165718229274337) (228 digits)
Using B1=3000000, B2=5706890290, polynomial Dickson(6), sigma=3741404180
Step 1 took 20467ms
Step 2 took 9969ms
********** Factor found in step 2: 16031381961952347637116191005607843989279
Found probable prime factor of 41 digits: 16031381961952347637116191005607843989279
Composite cofactor ((10^248+3)/(19*223*126165718229274337))/16031381961952347637116191005607843989279 has 188 digits
[/CODE]Group order: [URL="http://www.factordb.com/index.php?id=2"][COLOR=#000000]2^3[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=3"][COLOR=#000000]3^2[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=5"][COLOR=#000000]5[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=7"][COLOR=#000000]7[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=19"][COLOR=#000000]19^3[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=163"][COLOR=#000000]163[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=1307"][COLOR=#000000]1307[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=1487"][COLOR=#000000]1487[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=120163"][COLOR=#000000]120163[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=134639"][COLOR=#000000]134639[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=2337983"][COLOR=#000000]2337983[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=77402051"][COLOR=#000000]77402051[/COLOR][/URL]

[CODE]GMP-ECM 6.4.4 [configured with MPIR 2.6.0] [ECM]
Input number is (10^248+3)/(19*223*126165718229274337*16031381961952347637116191005607843989279) (188 digits)
Using B1=3000000, B2=5706890290, polynomial Dickson(6), sigma=1601899263
Step 1 took 14883ms
Step 2 took 8283ms
********** Factor found in step 2: 12824921391934305400334065366552991673187
Found probable prime factor of 41 digits: 12824921391934305400334065366552991673187
Probable prime cofactor ((10^248+3)/(19*223*126165718229274337*16031381961952347637116191005607843989279))/12824921391934305400334065366552991673187 has 147 digits
[/CODE]Group order: [URL="http://www.factordb.com/index.php?id=2"][COLOR=#000000]2^10[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=3"][COLOR=#000000]3^2[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=7"][COLOR=#000000]7[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=13"][COLOR=#000000]13[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=521"][COLOR=#000000]521[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=757"][COLOR=#000000]757[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=5209"][COLOR=#000000]5209[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=9883"][COLOR=#000000]9883[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=74699"][COLOR=#000000]74699[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=2923747"][COLOR=#000000]2923747[/COLOR][/URL] · [URL="http://www.factordb.com/index.php?id=3448573771"][COLOR=#000000]3448573771[/COLOR][/URL]

[QUOTE=Batalov;358988][CODE]Msieve v. 1.52 (SVN 886M)
Mon Nov 11 01:03:57 2013
random seeds: 187198dc e395e0be
factoring 9999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999992999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 (253 digits)
...initial square root is modulo 4585129
sqrtTime: 7150
p117 factor: 211857628677636264166676902114066256198845941934180788645905936104472665184004014226679863567582409810768749673482061
p137 factor: 47201510100993602326827859852217555163129145401663287219390072454758015933440888931227171966024510895342063068118093723212737563878094459
[/CODE][/QUOTE]

Impressive. How long did it take? How much processing power?

It was about 25 thousand hours for sieving, two hrs for filtering, 24 hours LA (on an 4x8 MPI grid), and two hours per sqrt.

Further exploration of pointless places
 

I ran 480 curves at 1e7 on {60..79}^512+1 over Christmas

No complete factorisations; a couple of new factors of interest:
[code]
e62:Found probable prime factor of 30 digits: 415279119367083281900859703297
e77:Found probable prime factor of 33 digits: 341126180420063151380968669975553
e65:Found probable prime factor of 37 digits: 1869269849997935174077690896845848577
e75:Found probable prime factor of 38 digits: 13734192372070026415774074593138282497
e65:Found probable prime factor of 43 digits: 4769997756860644904012186212092431977208833
[/code]

This was clearly untouched ground, I found a ten-digit factor of 79^512+1 that wasn't in factordb.

Reverse Smarandache numbers
 

Hi,

the last days I've factored 2 Reverse Smarandache type numbers for n=103 and n=104.

[URL=http://factordb.com/index.php?id=1100000000135133658]RSm(103)[/url].C160 = P53 * P108

Running yafu over night:
prp53 = 22633393225636817509048253413614523936779379142819839 (curve 50 stg2 B1=260000000 sigma=4172026601 thread=1)
Finished 400 curves using Lenstra ECM method on C160 input, B1=260M, B2=gmp-ecm Default

[URL=http://factordb.com/index.php?id=1100000000135133983]RSm(104)[/url].C149 = P52 * P97

Running msieve:
total time: 85.26 hours.
Intel64 Family 6 Model 58 Stepping 9, GenuineIntel
processors: 8, speed: 3.39GHz
Windows-7-6.1.7601-SP1
Running Python 2.7

Both reported to [URL=http://users.skynet.be/worldofnumbers/]World of numbers[/url].
Also shown on [URL=http://www.rieselprime.de/Others/Smarandache.htm]my page[/url].

[URL=http://factordb.com/index.php?id=1100000000135134178]Reverse Smarandache for n=105, C156[/url] factored in:

r1=505609049620430043564818948424594740095377638674786008583783558052966689 (pp72)
r2=1460218912197798897796479876892816487811802580775089126778648005904642208642833062339 (pp85)

rSm(106) C167 is factored

[CODE]GMP-ECM 6.2.3 [powered by GMP 4.3.1] [ECM]
Input number is 18177692096553830368675737725463580456289708131712261558393850692666532966863437168425047460718124572874681287411912149791448198810931545176347119222043777538034560927 (167 digits)
Using B1=110000000, B2=776278396540, polynomial Dickson(30), sigma=2318285213
Step 1 took 434355ms
Step 2 took 41233ms
********** Factor found in step 2: 414338872062791501547344020582712133249557
Found probable prime factor of 42 digits: 414338872062791501547344020582712133249557
Probable prime cofactor 43871558577296772025736976053227175068325706197701002055248304277569975777948248915189631633909304741312836729962564905149411 has 125 digits[/CODE]