sigma(303759912394367965087^6) = 1163 · 17137 · 122390927 · 496096189 · 18831851406578754695217341845201352652930437429509 · 34471494152587022328896671452463111550063183702701 (nice split!)
sigma(57094052065447889937556978631^4) = 5 · 1302645121 · 3464466041 · 73731834284894307940235489558454529334117671 · 6386692378652809581199493380527511855058671522389331

[quote=fivemack;207128]Am I reading the table wrong? 240492117224741725597 6 is a C96, an absolutely trivial GNFS exercise whether with a GPU or without (I'll do it)[/quote]

It was me misreading the table, I just saw GNFS and didn't notice how easy it was.

Some more results:
sigma(9736529984267551639^6)
r1=5739480900659408449168258743365016760801 (pp40)
r2=148440571848142141939306059170218369404474758751937636428702757643747734681 (pp75)

sigma(98419048214671022611^6)
r1=29241248973471249050868299761934879 (pp35)
r2=122897873042821021014220213486703206535051315107279460048964338764253969 (pp72)

sigma(234414539125735760881^6)
r1=380134099240072202189795800009753513821257591 (pp45)
r2=19803964014416723789084828002533307462816262923814583127084831107687677 (pp71)

sigma(252983942118045608479^6)
r1=4457203382991203296718216944667636474518343 (pp43)
r2=7217569338405762623220830618081021818034131795424332042510014589943822864123 (pp76)

I'll carry on until I'm counting days per number instead of numbers per day.

Chris K

I looked into using polynomial composition on some of the sigma(p^1) tasks and came up with a few candidates. I doubt the ones where sigma(p), p | sigma(q^2) will be all that useful, so I excluded them.

Script:
[CODE]grep " 1 " t1000.txt | perl -ane 'use Math::BigFloat; sub N {return Math::BigFloat->new(shift);} sub D {return R(log(shift)/log(10));} sub R {return int(shift() + 0.5);} $a = length($F[2]); $b = D($F[0]); $c = `/bin/grep " $F[0]\$" checkfacts.txt`; chomp($c); @d = split(/\s+/, $c); $e = (N($d[0])**$d[1]-1)/N($F[0]); if ($d[1] <= 13) {$e /= (N($d[0])-1);} if ($b < 1.4 * $a && $d[1] > 3 && log($e)/log(10) < $a/10) { print "sigma($F[0]) C$a D$b G".R($b/1.4)." poly($d[0]^".($d[1]-1).")+".$e->ffround(0)."\n";}'[/CODE]

Candidates:
[CODE]
sigma(1194867093367833983061432764872493628190644890583061564915554573142497236094355440948164731412150867740355566571664699858046287205086637) C106 D135 G96 poly(101011361988845464869763^6)+889
sigma(1326212461461076737644912847894367590445080229180268623521507921802776675184787410468458625994481389473320416301361061360983225375542808099323021668477) C122 D150 G107 poly(21329396889910742705842661^6)+71
sigma(879051940723580055746543425537999335624503926375906762793880782151995147077976874310316479726162482440382245000611654507344559443634007753352365121121) C130 D150 G107 poly(88040095945103834627376781^6)+529747
sigma(5614844584887175933279264011598835899447609690306443808036023846479091712722571376949385862729661419644748626851693977620690201365444390308003091198601) C131 D151 G108 poly(152407^30)+8391017142
sigma(22080435886626516945637217715046195689574340482513804855717661878648157232825906689028290929892754617405493168001014078597562620675870923944942026490460284567171205474668214601180997) C134 D181 G129 poly(1674947788493753179095247410947^6)+1
sigma(51271387494793330706583507866236355945525067098062719194472465813368489367086460533766299583794899292325114722050644173057750997664096442958821291007852297563869092881) C134 D167 G119 poly(26264382337^16)+26264382336
sigma(151610081666689684838973169400756206489325984857925348935766642066515543853165061342545623623497090163929181625445516576800704572122419370603008609850361358432819381605977) C135 D170 G121 poly(325220800093458627477824778607^6)+7804441
sigma(122049388824489340908655197898680249517095497203039058924709168561190690246126303614786835636432621266836796539179830161501239735096044158790041) C136 D143 G102 poly(56952152202269436602603340218578677109^4)+86199541
sigma(1107143537725947554658689059451849829090471252205614087123978608799086801542722844666199767976329692508527664624067804441860133113756105745980602226962031213112972265487621) C147 D171 G122 poly(5^262)+6094066806844
sigma(33967064818628105570428371821852312061168411783811040488642021266092993085361339253601985446861169567884880567515689118116178162598657497243307737178361) C147 D152 G109 poly(13^136)+12
sigma(90658648800638890796600037831317429016009500938092627207739565847129473006725393863778246303226213491004288300918034628632332994866147787733301467367279719049257) C149 D161 G115 poly(343632680481721^12)+29903655151189
sigma(98223050004178229506033385515949143838630827009681076200666460709092045729968111280182156756032104577679582925501330063883516957077344573778273155250568269812686441114503437) C158 D173 G124 poly(67925926254055524975061780331^6)+1
sigma(1252763643854396615424057223238912649030215010283484210574142027174772416111425664113983120903514963451470525049437388421091988243301472060308245302764730892284862185840224147636657451458107521509) C159 D195 G139 poly(132259604354473376342663326676479453^6)+4272619063673927
sigma(2718305123287994356831893002823904404818284041039062894254901450418025005195751093222391938716307391992763541354820941932752597535494831523979379711222726384674689926168933) C166 D171 G122 poly(687015720002749009^10)+8617247
sigma(2817966056096339458287311249389920340856018207587551050871178123804107396519044616257787459119723424650126229514077407232807401822324867173871413805248556408941979076772771793569734065353) C169 D186 G133 poly(3851^52)+3850
sigma(20948528256497991199338774610271336465959998203141735042999677579583173464660324977992088958272389261414212263232924590380261145065436526637367056499295185578601474237493176335234978586411900921) C181 D193 G138 poly(116856733421^18)+92101686155573780
sigma(272908017918779386485153313677657693500122526957618064783974325249242839033252145933854273430788742988707752260011441944670926204483275934631418030100269262430155178122303434328289935944517) C186 D188 G134 poly(2426789^30)+3162104764
sigma(20255984118348816247194023438463625929535007382958475379319909249397233089796323440859237528709329437893957511220150381211065498204696412510524404472750316619374987544762030737959621525626479718843849771053) C190 D205 G146 poly(961898407392288020436596589175811827^6)+39104063209
sigma(86942977836725015484888881444687486938457802109457831404277862954244495068925554562245904708914979777828437271208352807695577232003078373685800805657703325053357867976286032150299383307623991507471939306717694320918021) C209 D218 G156 poly(22406023^30)+8357446206
sigma(2846887948186840932838421028346212923389401024921184233982032869925012516940503551681650041497020176327547758752411813167995275054420061865210862199513191449678101944803876042290210555661910877669843762166319791802906005201) C223 D222 G159 poly(26010319^30)+26010318
[/CODE]

Sample:
[CODE]
sigma(39402408752184549336966124016297106825115166830903395209258983638370433385640142749538226756040422547837210718389901181701) C112 D122 G87 poly(2505420882950293453666143785857^4)+1
--poly
n: 1241291663152995841209502689982941481505644310559213003648302318581306591626152277827344010622180193281442455211
m: 2505420882950293453666143785857
c0: 2
c1: 1
c2: 1
c3: 1
c4: 1
type: snfs
skew: 1.19
--
prp44 factor: 55165930305860784439954094269474226420992769
prp68 factor: 22501055565832123832900019278099229945609167318140405598205428565419
[/CODE]

sigma(70841^36)
[CODE]prp76 factor: 3747115839410819881287722154872267936302096536400788414000178857228483282521
prp87 factor: 740652999976216489069105591347986323468093035854536074836616792798637204444015057857067
[/CODE]

[QUOTE=apocalypse;207301]I looked into using polynomial composition on some of the sigma(p^1) tasks and came up with a few candidates.[/QUOTE]
Nice.Thank you. I'll do sigma(13^136)+1.

Some more results:
sigma(48324254338285998810553779811^4)
r1=92865782264640318109709803606667094546033701 (pp44)
r2=3692081258940463779889036873252457414402495731475940305926168880741 (pp67)

sigma(57645352367255148283641834109^4)
r1=275093193135049153839284642441 (pp30)
r2=397425724600636055839820322095920291162849972560943417293907540620061026036780909041 (pp84)

sigma(61464952312244677403813855827^4)
r1=1584812542422131516779270840060929160891329091 (pp46)
r2=1150045631053067581292510890621954916378556865167163514503103048261 (pp67)

sigma(70185672354287301715822973599^4)
r1=17037225568324588403347887036024881 (pp35)
r2=19194683960497496733129258346868130566360881317132768504108751832960591 (pp71)

sigma(314756901565756922317^6)
r1=1593408429500253551301995940928385757698854861637437 (pp52)
r2=19312477690425173243057355671164072421296631590256216341706431 (pp62)

sigma(389886877017362551831^6)
r1=1167325862917728272878677537034494338332635166148101 (pp52)
r2=3009120634836656649655431472512528584262750168254305752648239470889755237 (pp73)

sigma(433471463064816310331^6)
r1=807969847413888459077618617524299994200981195741790569 (pp54)
r2=8058430795610571278465339811140611801264431314862899869 (pp55)

sigma(3869481297007413264839989^4)
r1=1661436307007172607644157586588271999821 (pp40)
r2=52895271672978518925566130056391431218515588599036816551 (pp56)

And reserving the next batch (I nearly ran out of work this afternoon):
sigma(445194303204046171141^6)
sigma(44611351^18)
sigma(604762571042373598801^6
sigma(128614489817758557898765676437^4)
sigma(161703035014964026334579729633^4)
sigma(165936770074941079041373815889^4)
sigma(20343393309397^10)

Chris K

Yet more results:
sigma(128614489817758557898765676437^4)
r1=721081181638284159573555834882376720427318431 (pp45)
r2=12240911040103795579906843864700642393963359910824110430387939553546861 (pp71)

sigma(161703035014964026334579729633^4)
r1=182384248310377202738700284420030911945271 (pp42)
r2=1868155402559234457844300850230116130316704032433919027624951 (pp61)

sigma(165936770074941079041373815889^4)
r1=8046164550212869478751020351278831 (pp34)
r2=14867205669788181357824854353747250171071891730465512299068646793601 (pp68)

sigma(129994124149^12)
r1=209169685249137871139635995280414261100425901 (pp45)
r2=7771405933927854731005728663820782362862485640733417 (pp52)

sigma(20343393309397^10)
r1=5707446798945301546945573441143443602765002139 (pp46)
r2=93996808856663616204874016720183894344492310434589 (pp50)

And reserving some more:
sigma(369023021^16)
sigma(48552947^18)
sigma(653058596067301423729^6)
sigma(1003947793406659309063^6)
sigma(1142088053833817665537^6)
sigma(268760258883389029849216937779^4)
sigma(283621533437336511818240941069^4)

I've noticed that the quartics and sextics seem faster than higher degrees of the same SNFS difficulty.

Edit: Worked out why quartics and sextics are faster, apocalypse's perl one liner to work out how hard they are calculates how many digits a^(b+1) has for sigma(a^b). But for quartics and sextics the SNFS difficulty is actually the number of digits in a^b.

Chris K

[QUOTE=chris2be8;207460]
I've noticed that the quartics and sextics seem faster than higher degrees of the same SNFS difficulty.

Edit: Worked out why quartics and sextics are faster, apocalypse's perl one liner to work out how hard they are calculates how many digits a^(b+1) has for sigma(a^b). But for quartics and sextics the SNFS difficulty is actually the number of digits in a^b.
[/QUOTE]

Difficulty is certainly part of it.
[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.

Another thing I learned (from fivemack in the thread "Running GGNFS") after I wrote the difficulty script is that it is also important (perhaps more so) to keep the evaluations of the two polynomials small so they are more likely to be smooth in the factor base to maximize the number of relations yielded.

Using sigma(48552947^18) as an example, factoring (48552947^19 - 1) as
[CODE]p(x,y) = 48552947*x^6 - 1*y^6, q(x,y) = 1*x - 48552947^3*y[/CODE] with typical sieving values k[SUB]x[/SUB],k[SUB]y[/SUB] ~= 10^7 will give p(k[SUB]x[/SUB],k[SUB]y[/SUB]) ~= 10^50, q(k[SUB]x[/SUB],k[SUB]y[/SUB]) ~= 10^30.

On the other hand, factoring (48552947^20 - 48552947) as
[CODE]p(x,y) = 1*x^5 - 48552947*y^5, q(x,y) = 1*x - 48552947^4*y[/CODE] with typical sieving values k[SUB]x[/SUB],k[SUB]y[/SUB] ~= 10^7 will give p(k[SUB]x[/SUB],k[SUB]y[/SUB]) ~= 10^43, q(k[SUB]x[/SUB],k[SUB]y[/SUB]) ~= 10^38.

The reduction of the larger value from 10^50 to 10^43 should probably more than make up for the increase in difficulty from 146 to 153. In general polynomials with larger coefficients are less likely to evaluate to numbers smooth in the factor base and therefore will yield fewer relations and require more effort to factorize.

sigma(3169^52)
[CODE]prp64 factor: 1709750862450097630029730418637977790700641627772053309975927719
prp109 factor: 4076664426531199154509447796199785974599280164338933274532863603671840997005973725995359959441518165420315303
[/CODE]

I'm going to do some ECM on the C225 from sigma(547^88) in the roadblocks file next, probably a few thousand curves with B1=11e7. I'll try to keep the factordb page up to date & I'll post back here when I'm done.

Having recalculated SNFS difficulties I'm reserving the following:
sigma(1669983316252662188125989563369^4)
sigma(2121379629902041750864180018843^4)
sigma(3110504728150197490617159721597^4)
sigma(386467134920697428403287941271^4)
sigma(4664607259009421338832033924593^4)
sigma(4704529625339975266939694199859^4)
sigma(5035815400597284456708244650043^4)
sigma(538454721781787447477597948423^4)
sigma(568944611281729293283376872939^4)
sigma(6503968953043892440658971030483^4)
sigma(8379347128335604166268202471087^4)

Ie everything up to SNFS difficulty 124. They should all be reasonably fast.

And here's the latest result
sigma(445194303204046171141^6)
r1=1390321640475112813286259957154877730563 (pp40)
r2=5599926252067404645186544914300120796865906450656803712787680312543002861345054079769 (pp85)

sigma(44611351^18) is taking a while to do. I should have a result tomorrow.

I've noticed a few lines in the latest t1000.txt with exponent 1. I assume they will all have to be done by GNFS.

Chris K

[QUOTE=chris2be8;207559]I've noticed a few lines in the latest t1000.txt with exponent 1. I assume they will all have to be done by GNFS.[/QUOTE]

See post 201 in this thread. Composition from backtracking sometimes works for these.