Reserving the first few GNFS targets:
sigma(446632525031622232178391539765730794032997038685577008762353605301524822751762699590186359593^1)
sigma(233737539059460333409923964178251456016552547427^2)
sigma(327630961466779424810385652526155864807281857263^2)
sigma(2163254010885746235845437816794637833999858626522574858951227819629744305616126667345649124972001377^1)
sigma(128913780332845744550757714811875858406723998870419^2)
sigma(182080264391977687153394142116599354871427372364577^2)
sigma(210302379034799710595960865883327736284685168485033^2)
sigma(325494807812587105689689342848972158212035828985191^2)
sigma(44323579858834662206799254307759703799930741548496735923963496143519404379504784662230835584392694361^1)
sigma(1084588588364016850642374523063363198062560323249351^2)
sigma(579076586454227273501855189738081169271292153432899^2)

Chris K

[quote=apocalypse;207529] [code]D ~= log[sub]10[/sub](a^b) for b in 4,6,10,12 because they work for (x^(b+1) - 1)/(x-1)
~= log[sub]10[/sub](a^(b+1)) for poly (a*x^b - 1)
~= log[sub]10[/sub](a^(b+2)) for poly (x^(b+2) - a)[/code]etc.

[/quote]

Here is my script to list everything sorted by equivalent difficulty: [code] perl -ane 'chomp($F[2]); $c = length($F[2]); $d = log($F[0])/log(10) * ($F[1]); if ($F[1] >12) { $d = log($F[0])/log(10) * (1 + $F[1]);} $d = int($d + 0.5); $g = int($d / 1.4 + 0.5); $t = "GNFS";$diff = $c; $suffix=".n"; if (($c > $g) and ($F[1] >3)) { $t = "SNFS";$diff = $g;$suffix=".poly" } print "$diff o$F[0]_$F[1]$suffix sigma($F[0]^$F[1]) C$c, D$d, G$g, $t\n";' t1000.txt | sort -n [/code]And a few more results:
sigma(44611351^18)
r1=1605338393162533971858815730908351445692996675475654035707 (pp58)
r2=305105598427050205685857680046231692723766079130149461931050258453343591420793067 (pp81)

sigma(604762571042373598801^6)
r1=11587406965808689412035036162479096459537404172898741 (pp53)
r2=15720065434550263176080279023379407087822165844122306563343505832251 (pp68)

sigma(1084588588364016850642374523063363198062560323249351^2)
r1=26510827725432172521233808897055841043852241 (pp44)
r2=14790590699927237970403950982620046341390250364380200474211 (pp59)

sigma(128913780332845744550757714811875858406723998870419^2)
r1=29894215075150109410304441962753933081 (pp38)
r2=3540885480148424362589137831006836302980168966065195539785993 (pp61)

sigma(182080264391977687153394142116599354871427372364577^2)
r1=223428711342303434268671417812097310633447931 (pp45)
r2=2032656110436488900743630232419935728661512536854536689 (pp55)

sigma(210302379034799710595960865883327736284685168485033^2)
r1=93139847964516074577466843384162002651653779 (pp44)
r2=12175540218240085524510235874410786378193781638656983583 (pp56)

And reserving a few more:
sigma(17049927741295723419455438456527^4)
sigma(25865202218735920969913319342977^4)
sigma(31503547855818058003443589783807^4)
sigma(39091097830253216170021088927077^4)
sigma(102206271707100588285946234912543^4)
sigma(105486612030108129703082913595759^4)
sigma(1213312309461502477471^6)
sigma(1573246909647668001337^6)
sigma(1613833928517310528571^6)
sigma(1953340318346474862709^6)
sigma(2395419413743258360243^6)

Chris K

[QUOTE=chris2be8;207677]Here is my script to list everything sorted by equivalent difficulty: [code] perl -ane 'chomp($F[2]); $c = length($F[2]); $d = log($F[0])/log(10) * ($F[1]); if ($F[1] >12) { $d = log($F[0])/log(10) * (1 + $F[1]);} $d = int($d + 0.5); $g = int($d / 1.4 + 0.5); $t = "GNFS";$diff = $c; $suffix=".n"; if (($c > $g) and ($F[1] >3)) { $t = "SNFS";$diff = $g;$suffix=".poly" } print "$diff o$F[0]_$F[1]$suffix sigma($F[0]^$F[1]) C$c, D$d, G$g, $t\n";' t1000.txt | sort -n [/code][/QUOTE]

The tricky part is that the difficulty is of the polynomial, not of the number per se, so in certain cases (e.g. when $F[1] doesn't divide 4, 5, or 6) you would almost certainly factor the higher difficulty x^(b+2) - a, as well as in more subtle cases (e.g. where the higher difficulty polynomial has much higher yield).

I addressed the first case in my initial difficulty script with:
[CODE]$d = log($F[0])/log(10) * (1 + $F[1]); if ($F[1]%4 != 0 && $F[1]%5 != 0 && $F[1]%6 != 0) { $d = log($F[0])/log(10) * (2 + $F[1]); } elsif ($F[1] >= 4 && $F[1] <= 12) { $d = log($F[0])/log(10) * $F[1]; }[/CODE]

I haven't yet scripted up a nice heuristic for the second part (or to build in polynomial composition). I have some thoughts, but I need to see how they work in practice, since I'm still a little behind on the theory :smile:

I'm only interested in the low end at present since I'm working through the easiest cases. In practice they are almost all degree 1, 2, 4, 6, 10 or 12 so my script works well enough.

Relative speed of SNFS and GNFS is only approximate though, it depends on how good a GNFS poly you find.

Talking of which here are my latest batch of results.
sigma(268760258883389029849216937779^4)
r1=1409700368859390535640400000291055676255229971 (pp46)
r2=5232560790174748769213850125620987728664672317207419890258826411 (pp64)

sigma(283621533437336511818240941069^4)
r1=2244387182602163245885505218437032224475246949431 (pp49)
r2=20602954351633520875970832102846201134622704700552731 (pp53)

sigma(1669983316252662188125989563369^4)
r1=8504123927468857480972673121718435024316703881 (pp46)
r2=52686715275653858963898129029423411768223634242937480261 (pp56)

sigma(2121379629902041750864180018843^4)
r1=1129418316864791282932549165013581 (pp34)
r2=101346860421103344293379525364269327679942154506419716897130314498109855470852581 (pp81)

sigma(1003947793406659309063^6)
r1=3133638801056710964307127151572449324150110759 (pp46)
r2=9302459143253883608700045515828210004138505897448155881637 (pp58)

sigma(1142088053833817665537^6)
r1=265260921620887064937771089659306241 (pp36)
r2=5103443471077055728360531169216020367634168446725725791413259045273 (pp67)

sigma(2163254010885746235845437816794637833999858626522574858951227819629744305616126667345649124972001377^1)
r1=8234538025576891351663146981928223015417239 (pp43)
r2=131352481715827283122099191124412672476001395658247218551 (pp57)

sigma(233737539059460333409923964178251456016552547427^2)
r1=670706671921370387925345087804828580668194401 (pp45)
r2=27152076783464399866971400096709402889335120142319 (pp50)

sigma(325494807812587105689689342848972158212035828985191^2)
r1=16805271165656689114894704043987527901538689 (pp44)
r2=2101461080645268205118387826485483252182891720052670319219 (pp58)

Chris K

PS. How long does it take for a session to time out? This is the second time I've had to log on again after taking some time making sure I had the right factors to go with each number.

Thanks to [boinc.at] jbs at yoyo@home,

2990127956922054333455948401313838513991 | sigma(127^210)

Leaving a 403 digit composite cofactor.

Post 130 in this thread has been updated.

191^103-1 sieved (100.8 million relations, 89M or so unique); started linalg.

Don't know

a) whether I'll get factors before I leave the country on Saturday afternoon
b) whether it's feasible to ssh into my machines from a netcafe in a swamp in Botswana

so factors will arrive either around Saturday or at the beginning of April.

A few more results (things seem to be getting slower):
sigma(327630961466779424810385652526155864807281857263^2)
r1=22892381895129525500144028799527149476411 (pp41)
r2=1562995168776851024662896233704556085043614579200642401 (pp55)

sigma(44323579858834662206799254307759703799930741548496735923963496143519404379504784662230835584392694361^1)
r1=16243517830472440929605239920238531 (pp35)
r2=59319421688330496648817157264162978727547954044761690935519872137 (pp65)

sigma(446632525031622232178391539765730794032997038685577008762353605301524822751762699590186359593^1)
r1=331429867496313712351764583075980224291 (pp39)
r2=673796433021519774640310836288727483223961016051515367 (pp54)

sigma(3110504728150197490617159721597^4)
r1=474638501683939998050877669504101676071 (pp39)
r2=151159236780857278085914407418622383826844999498060628010330870723329781 (pp72)

Chris K

Some more results:
sigma(579076586454227273501855189738081169271292153432899^2)
r1=100030510236461518420622229050084914255077463 (pp45)
r2=1117424714342308560988277701735700056963187393973665370409 (pp58)

o386467134920697428403287941271_4
r1=4700190232754902982191783828240571 (pp34)
r2=42920955205555784183986901174087861852543595606512740370261271454801861 (pp71)

sigma(4664607259009421338832033924593^4)
r1=36386160992963090593337932048885935492075059583361 (pp50)
r2=1149159210160267871992984438979626624771029413837849776377197191 (pp64)

sigma(4704529625339975266939694199859^4)
r1=31658889308093350260832038469747903691 (pp38)
r2=1486251603719353659651981019094024751056289879520457158850741 (pp61)

And reserving another batch, having checked against the latest t1000.txt and checkfacts.txt:
sigma(55350266314895595867025212145997^4)
sigma(8058430795610571278465339811140611801264431314862899869^2)
sigma(83172085527497335812994589511133^4)
sigma(142684958271631124501863184377609^4)
sigma(157498899221694152189827767885989^4)
sigma(161623259620322415732121172922331^4)
sigma(165978531967050794414709318599449^4)

Chris K

Some more:
sigma(538454721781787447477597948423^4)
r1=8193912944867468250381149902384220179032079556431 (pp49)
r2=1684234202404432767010094290612133136416815396700320270831912831 (pp64)

sigma(5035815400597284456708244650043^4)
r1=25996809969240756582665313987873840798773303480431 (pp50)
r2=12679492690412590656516312119373238102631502674048537710689300566845721 (pp71)

sigma(102206271707100588285946234912543^4)
r1=50152942772563234068159408312234759459961 (pp41)
r2=97368435099686679902595569299801161867918200470963610711774011 (pp62)

sigma(105486612030108129703082913595759^4)
r1=302082424207988233684766069881 (pp30)
r2=14781075861560230530968539444069997158999549551612302215257388605680421788851739578441579591 (pp92)

I would have had more, but I ran out of disk space overnight.

Chris K

from [URL="http://www.lri.fr/~ochem/opn/t600.txt"]t600[/URL]

sigma(970244551306876240403^10) =

3747469240420563448461930575471274228413389823 x C165

In before the bell ...
 

191^103-1 is finished, and not an ECM miss even by current standards:

Fri Mar 12 15:44:34 2010 prp115 factor: 8742480382747760307000631794005028106590299386513617942966930026414320584721472504488525411610859151381075605984733
Fri Mar 12 15:44:34 2010 prp118 factor: 5321677583871343234090935878718828198568472674649204112537448734063507775483494418871698489215844672230924537926467381

6600 CPU-hours of sieving (took about two weeks, roughly one CPU-day per hour, using my rather-too-large home cluster); with four threads on i7/2800,

Tue Mar 9 09:26:39 2010 found 14424532 hash collisions in 100816295 relations
Tue Mar 9 09:42:17 2010 begin with 89585556 relations and 80988402 unique ideals
Tue Mar 9 10:09:42 2010 weight of 6958082 cycles is about 487384167 (70.05/cycle)
Tue Mar 9 10:10:51 2010 RelProcTime: 3464
Tue Mar 9 10:22:15 2010 matrix is 6956557 x 6956782 (1986.7 MB) with weight 486624276 (69.95/col)
Tue Mar 9 10:22:15 2010 sparse part has weight 451240188 (64.86/col)
Fri Mar 12 06:08:31 2010 BLanczosTime: 244660
Success on the fifth square root, so
Fri Mar 12 15:44:34 2010 sqrtTime: 34563