[quote=wblipp;206886]It might be worth back tracking. For example, the first case of an exponent 2 in t500.txt is for base

5502997992185822133325044025114218231630382402194049024447

Looking in checkfacts.txt, we see that this prime is most of

sigma(1785003035704867089959419277971^2)

and 1785003035704867089959419277971 is itself most of

sigma(27282710017315537^2)

and 27282710017315537 is a pretty large factor of

sigma(108541^6)

So you might find good SNFS polynomials by re-expressing in terms of one of these back-tracked bases.

William[/quote]
the poly for sigma(5502997992185822133325044025114218231630382402194049024447^2) is:
x^4+2*x^3+582*x^2+581*x+335821
took me ages to work out so i may as well post it:smile:
i first worked out the poly for factoring sigma(1785003035704867089959419277971^2) first by mistake and then tried converting my method to the proper number
it turned out that sigma(27282710017315537^2) is divisble by (3*1[COLOR=Red]39[/COLOR]) and sigma(1785003035704867089959419277971^2) is divisible by (3*1[COLOR=Red]93[/COLOR])
i failed to notice that the highlighted digits had swapped:blush: that wasted a lot of time:smile:

[QUOTE=henryzz;206899]the poly for sigma(5502997992185822133325044025114218231630382402194049024447^2) is:
x^4+2*x^3+582*x^2+581*x+335821
[/QUOTE]

I went ahead and ran that one:
[CODE]prp47 factor: 70489496535231776758399674369130692330187659043
prp50 factor: 81194777220024677651378678180030604682993282219357[/CODE]

Unfortunately it looks like there is only one other sigma(p^2) where p | sigma(q^2) chain in t500.txt, and from a cursory inspection, it looks like most of the sigma(q^n) values have too many factors to make

[CODE]p(x) = a^2 f(g(x)/a)
where
f(x) = x^2 + x + 1
g(x) = (x^(n+1) - 1)[/CODE]
practical.

I wrote a "one-liner" to get the log[SUB]10[/SUB](a) for all sigma(p^2)
[CODE]grep " 2 " t500.txt | perl -ane 'if ($F[1] == 2) {$a = `/bin/grep " $F[0]\$" checkfacts.txt`; chomp($a); if ($a =~ /^(\d+ \d+) (\d+)$/) {$b = $1; $c = $2; $d = `/bin/grep "^$b " checkfacts.txt`; @e = split(/\n/, $d); $f = 0; if ($b =~ /(\d+) (\d+)/ && $2 != 3) { $f = log($1)/log(10); } for $e (@e) { @g = split(/\s+/, $e); if ($g[2] != $c) {$f += log($g[2])/log(10);}} print "$F[0]\t$f\n"}}'
[/CODE]

which gives:
[CODE]5502997992185822133325044025114218231630382402194049024447 2.76267856372744
12806205108261614728117250368818758471115817601449128327311 75.3630783943435
4828882010768253255625697644885148701419507806262733969 21.5594285807886
36805908914339368365177508331329628145229111570447371537821 33.1850075415384
67300960941973270216603743178608534313917950387824769083 62.42475704922
5401453999912054294903561906118030238109101698398612191623375347 17.231125719338
84914530903304553545666061499881394732390301127632672185279 19.9745509178073
5775464745322578520822954101491045929218032858547412479330389 41.8080684037822
69578974582861797141313609380633155003252180631220112754924473709869 25.4875726888902
20267747136615386578739143471095264354508796953272082700055339846745091 36.9980337791246
148467031893746524227485026611653806025492334017387237830406481156482009 36.0392180005993
505721798544031140740638061509529267073352657243267449823387256782461164389882661 11.0416349970254
98138549979439469138489095158515442021127198477956313572191080279412141989 32.3098606843655
172918880358232491224830637233905569874719873338729915557207194422824133751 20.7349139655763
227852513819803851204263653623117987276133170184156001598484507703001831869 31.4474636421946
385254252308931745111497517354638378475339965873128329064640766446490856069184751 38.7943902612992
12577647923565067982570531255067267090413921105116610018487966576192735681081030635731 15.9454161374321
321381569252585866953628783126948367071187906389518216907098417372109834635071531 16.154488650764
461868914689731864111389613832292217983771315608257314290991856316645179572881366073 20.7085173559131
42223362249943191270479011547514088211349705097809873373648793992949125900561049993 29.6638031545409
903344592651066639838492820709976987862568330541660232234109502106802059231851745003701 26.5973419181551
[/CODE]

The next most useful polynomial looks to be derived from
[CODE]$ grep "^911 31" checkfacts.txt
911 31 683
911 31 176887
911 31 505721798544031140740638061509529267073352657243267449823387256782461164389882661[/CODE]

which gives
[CODE]
? f(x) = x^2 + x + 1
? a = 683 * 176887 * 910
%1 = 109940577110
? a^2 * f((911*x^30-1)/a)
%2 = 829921*x^60 + 100155865745388*x^30 + 12086930495169915374991
[/CODE]

but the coefficients are already unwieldy and I don't think this is better than GNFS for the remaining cofactor. Unless I've made an error, I don't think this technique will be generally useful for the sigma(p^2) tasks in this project.

It looks like there are 43 remaining tasks in t1000.txt with a < 10[SUP]5[/SUP]

[CODE]100860848582065425635519047877077112241482572096184039107 2.43616264704076
356396319807932416808478086958280576033087391279487445655878686015919 0.477121254719662
80938053859323924441034825605861453767680358105990761053090679956450783212695493212057 0
33736677531039915642812529579891149374692367272008324878753 1.27875360095283
336756508711114813210525750593533959832184338890841231366505661 0.477121254719662
8723064423943200254410958233775487298905572251907348668396543541017 2.92890769024395
244843759373522464138800829966383113629900505470123750552624284061 0
1688527160526451692513197069042282970284044468045537666230082083397478785508463519991 0
13012432053809145115622094807998190254812633756766651753844794421589559232065341823005248594539611 0
1649476056504028070856939440126046115057895951345037976424836003762477630761301549 1.96848294855394
1606705298255981044159575916287625882510186775459893939942675228291 1.75587485567249
29071023748327672813409243996652769965223372219693277145946195914872139961 0.477121254719662
59086113407433947521317906400603315589816488591785334607842447680087857299 4.71889179736579
155234288670052021011894206983471668817109570595909949873047496511758964659 4.64233589462999
94319186834034012396785356863537151198151266836804837268335875274335961468833997366886655872213 1.5910646070265
85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457 0.845098040014257
889879764780235158902477203791385111811056969337198626984987539808659570446311 1.11394335230684
13144835032641767610904893692004408713494636253315836302858985835167220744482345766031056713 2.68930885912362
772580091968503207075530910318688563679895158160480207121434026938937432895564736785949 2.8992731873176
621577855103779232020354854430799478674267898455633513341518743986760647186871128363 4.64433024853358
792705753533285460764196627044658731054221065976124160365993127009298720316792974934389441 0.477121254719662
671128236077515738032837904111506686388860348391579216895155547781469320228005072453696196476372861 1.11394335230684
44245803464617839294734385296575692370533096490259020719500329152733942767235009936739279318759677 1.32221929473392
589934943146977358794812334946703260891667832185594287986281028345953880265717988267466392663082933477 2.95568775031351
3741648904975135735492021626834234241649645101427624598719320614397465651449376471929741110089690453 4.34097921660363
609291420723967102569046934704073009248140868144164845615377717881353376253507216236272594333549529418601089 1.56820172406699
39336521495350642918410403933286526179722603250101581703275499647659136336492611036653237376347 4.52295256906742
285980111490689789770327491263171820649510345474079812267515979364881902439633649782777909508034007614477624565657 0.477121254719662
39189786792441251178275531624458156105983483780150360330006489750437945287388018043997228846569559721750538278568937 2.97266559226611
354663404509497202680421171823973363403715814430877142068022809134014434055952184581764676465938617706521 3.73391915101239
37456476993611934910942884079251563764980037287112723737330594708916619660838080324694432246175301248638931429 3.9739125704197
93379858651474679531202595347950618968237427956330582055651589394217164172715912778101206658341568416813557 0
625480301180450039142955317616449489404291013867114950182389548709202430829836083882948048338149055879055477 3.73407940728059
37888683035768375759970187456398397758136601005129717609077596435167199656624622763755864366726196385367718271 1.49136169383427
13750621423269977545797375664534485249208795949678921216923216430349049728637490407903242134865022017278746237742464286632890576385130257 2.959518376973
28766732153500846964897566503296979358475928181733147562692457239964971309395783134963292555889164273935636598772687925169 0.477121254719662
522479742689128449814000772046374748429821256832559929066972801408204225569220325701543550281394390498590839663937 0.477121254719662
3727029213159221227481880371758988192479699585560736992193446518019059007305528138257031138276615655165685013865802400046939654576178541 1.27875360095283
2557979250062842440919959580951855364625644742139230192496743270345842062500486270705748676162909259636192241002845021368830236136757 2.17897694729317
567000232521795739625828281267171344486805385881217575081149660163046217465544573355710592079769932651989153833612198334843467861091902034340949 1.04139268515822
3364134212814498888636588437428925181558554547018868067559465096021275448060742344390755257527049173609544578675371857565876968887946124269533307 0
73004735964705042068814643307796564935889091565596520218261053278055679104290197628217951566809162828702260699073627697676660026688517389942561844507382131 4.03818286599066
33967064818628105570428371821852312061168411783811040488642021266092993085361339253601985446861169567884880567515689118116178162598657497243307737178361 1.11394335230684
[/CODE]

[QUOTE=apocalypse;206898]How much ecm has been done against the roadblock numbers? I'd like to work on them, but I don't know where to start.[/QUOTE]

See post #130 in this thread. I'm keeping it updated with all the ecm work I know about.

[url]http://www.mersenneforum.org/showpost.php?p=202991&postcount=130[/url]

More results:

sigma(126860821^16):
r1=213505995736244544845260255355444673247 (pp39)
r2=43227959127300286150369050288914721828527 (pp41)
r3=11186985970678062138497875424367540433275907 (pp44)

sigma(145406578159655383969^6)
r1=3824737102301952029620348659651099467730079 (pp43)
r2=89611557051666659001554180249927744821490712912549602939193288810959 (pp68)

sigma(154734729607724347493^6)
r1=8989138036540531185253000923215228046700634654431201 (pp52)
r2=7992615802404778942285635622648092818556693969590082457 (pp55)

The process for sigma(p^2) looks a bit too complicated to script up. So I'll leave them to other interested parties.

Chris K

More results:
sigma(4689632444309897463803449841^4)
r1=83647721755806543427503678873367323472630027563181 (pp50)
r2=34297367061272061616363392530709772535056988328656641 (pp53)

sigma(5342436683061058354198492453^4)
r1=3217171681420402252392905680992640689425935739431 (pp49)
r2=8168110491133361461091059827756770787349984020753242068459061 (pp61)

sigma(7780457401386385640670987149^4)
r1=530444960325678016197918719294491 (pp33)
r2=52736207413364818821718705863430635841270007461938736247523235484795628232781 (pp77)

sigma(581476728991^12) (I think I copied the wrong poly to the dir where my script runs, which is why I didn't reserve it)
r1=2221327652541825903274501323462374725507 (pp40)
r2=2066249961459750431644625892243824089902484310026561378036353513096974437 (pp73)

And reserving:
sigma(389886877017362551831^6)
sigma(433471463064816310331^6)
sigma(57645352367255148283641834109^4)
sigma(61464952312244677403813855827^4)
sigma(70185672354287301715822973599^4)

I'll leave the following GNFS targets to someone with a GPU for polynomial search:
sigma(240492117224741725597^6)
sigma(303759912394367965087^6)
sigma(57094052065447889937556978631^4)

Chris K

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=chris2be8;207126]sigma(303759912394367965087^6)
sigma(57094052065447889937556978631^4)[/QUOTE]

These are similarly easy. I'll do them.

A SNFS result:
[CODE]sigma(11807^42)
prp60 factor: 598910770964931201718652782199984034907876410474921125717077
prp101 factor: 37676392438412411375059458909903909316832718328095093450425530615119929633311904520436458635600825633
[/CODE]

Some ECM results
[CODE]sigma(25125200742702138527^16)
c281 = 2119516645339138079890144425812969 * c247

sigma(172748175775271914845416281805553997185490455536821^6)
c284 = 123921681027330918924612197 * 2672563430311978558236477521857 * c227[/CODE]

I wrote a (increasingly misnamed) one-line script to generate poly files for functionally composed sigma(p^2) values.
[CODE]perl -ane 'use bignum; sub D {return int(log(shift(@_))/log(10) + 0.5);} if ($F[1] == 2) {$a = `/bin/grep " $F[0]\$" checkfacts.txt`; chomp($a); @b = split(/\s+/, $a); $c = `/bin/grep "^$b[0] $b[1] " checkfacts.txt`; @d = split(/\n/, $c); $e = 1; if ($b[1] != 3) { $e = $b[0] - 1; } for $d (@d) { @f =split(/\s+/, $d); if ($f[2] != $b[2]) { $e *= $f[2];}} if ($b[1] == 3) { $g = $b[0]**(2*($b[1]-1)); print "#C".D($F[2])." D".D($g)." G".D($g**(1/1.4))."\ntype: snfs\nn: $F[2]\nm: $b[0]\nc0: ".($e*($e+1)+1)."\nc1: ".($e+2)."\nc2: ".($e+3)."\nc3: 2\nc4: 1\nskew: ".(($e*($e+1)+1)**0.25)."\n";} else {$g = $b[0]**(2*$b[1]); $h = 2; if (($b[1]-1)%3 == 0) {$h=3;} print "#C".D($F[2])." D".D($g)." G".D($g**(1/1.4))."\ntype: snfs\nn: $F[2]\nm: ".$b[0]**(($b[1]-1)/$h)."\nc0: ".($e*($e-1)+1)."\nc".$h.": ".($b[0]*($e-2))."\nc".(2*$h).": ".($b[0]*$b[0])."\nskew: ".((($e*($e-1)+1)/($b[0]**2))**(1.0/(2*$h)))."\n";}}'[/CODE]

The "use bignum;" makes it pretty slow. It could be sped up with more judicious use of multiple-precision arithmetic.

[CODE]
# 100860848582065425635519047877077112241482572096184039107 | sigma(165936770074941079041373815889^2)
sigma(100860848582065425635519047877077112241482572096184039107^2)
poly:
#C108 D117 G83
type: snfs
n: 895423886690813236595276815273932699969294991769532806541199763388502776044173947098938243755544261863664237
m: 165936770074941079041373815889
c0: 74803
c1: 275
c2: 276
c3: 2
c4: 1
skew: 16.53787686227397334882784874743856953888
--
prp42 factor: 390107233242397287356015618006603288797963
prp67 factor: 2295327567367693643024213504735796771918311722153151284381043479399
[/CODE]

Some sample polys (which I don't intend to run) showing that the script works when the previous sigma(p^n) in the chain has n!=2
[CODE]
# 354663404509497202680421171823973363403715814430877142068022809134014434055952184581764676465938617706521 | sigma(5419^28)
sigma(354663404509497202680421171823973363403715814430877142068022809134014434055952184581764676465938617706521^2)
#C198 D217 G155
type: snfs
n: 1853850230723632249136921930341024637904473174796123981355911899141339467647426658257965662420965831050171314381836328397016461769476177816266078119193669622205965589493810119034728721473915294386131
m: 18830771529114369899502254666523001780868148423704521
c0: 29349307
c2: 29349304
c4: 29365561
skew: 0.999861594889637693360213620650932410697

# 567000232521795739625828281267171344486805385881217575081149660163046217465544573355710592079769932651989153833612198334843467861091902034340949 | sigma(11^138)
sigma(567000232521795739625828281267171344486805385881217575081149660163046217465544573355710592079769932651989153833612198334843467861091902034340949^2)
#C267 D290 G207
type: snfs
n: 2392198126144608273513479610147620749531901159220211414582939088551950693781693454625203188089712730329446954923786707279288008217609646426090545940363898995048177189986547178214547288000420885602286885151370266704251670535660870827913792986501394383246190031072331263
m: 801795320536133573571931534665380233173841533961
c0: 91
c3: 88
c6: 121
skew: 0.9536214335048881035541177534133425577741[/CODE]

I don't know if these are actually useful polynomials, but they seem to be better than GNFS would get.

the divisor of sigma(240492117224741725597^6) splits with

prp38 factor: 96022106584004884729501051826021586269
prp58 factor: 1710011547381748249982052806907419640310143585338596863529