Polynomial selection for 2,1109+ c225

[R.D. Silverman]

Originally Posted by frmky

Likely since the e-score is an integral combining contributions from both size and root optimization. msieve reports both the final norm characterizing the size and alpha characterizing the root properties of the polynomials. Attached are histograms of each of these separately for the same 2 million polynomials.

The alpha pdf does seem to have a longer tail to the right.

[frmky]

Originally Posted by R.D. Silverman

The alpha pdf does seem to have a longer tail to the right.

Yes, but also that's the worse side. Better root properties correspond to the left side of the distribution.

[Gimarel]

A few better polys:

n: 126451876805119252959661548967232013601866431183534308908427174011662073024932261126233275630388431500884665768427172907593022873289431612806891303891687570778305960728323476541491713906981621716831593952842244282430058291761
# norm 4.071969e-16 alpha -9.752538 e 2.347e-16 rroots 4
skew: 4099972.17
c0: 75411469889145395046741594539510303818011021471
c1: -19691135073162315084594784624103449198750
c2: -99438233376176899175468255217094130
c3: 4394174158803947144118894880
c4: 16400189169252108563099
c5: -232506169843650
c6: 4347000
Y0: -2754045785939680784737571685756365496
Y1: 4538373177770229797351
# MurphyF (Bf=6.872e+10,Bg=3.436e+10,area=1.766e+18) = 2.477e-09

n: 126451876805119252959661548967232013601866431183534308908427174011662073024932261126233275630388431500884665768427172907593022873289431612806891303891687570778305960728323476541491713906981621716831593952842244282430058291761
# norm 4.143131e-16 alpha -9.942142 e 2.339e-16 rroots 6
skew: 45290543.52
c0: -269505104302737992226579881824069273348411150390400
c1: 31592047037538685851862704375760833455315760
c2: 1703133447582614942920906384956167268
c3: -129971466498994600011942115252
c4: -2314332759744553531033
c5: 29445952854202
c6: 50400
Y0: -3686227314243652116010167240943024263
Y1: 30016011206759833401211
# MurphyF (Bf=6.872e+10,Bg=3.436e+10,area=1.766e+18) = 2.466e-09

n: 126451876805119252959661548967232013601866431183534308908427174011662073024932261126233275630388431500884665768427172907593022873289431612806891303891687570778305960728323476541491713906981621716831593952842244282430058291761
# norm 4.031559e-16 alpha -12.133288 e 2.292e-16 rroots 4
skew: 29346738.87
c0: 59455777517328054184793219662976360401235607337335
c1: 112779972076226040399992792530106515604140047
c2: -4298413822937233372631606035726816679
c3: -1413322411017627466877645728827
c4: -821206219249483879936
c5: 180781205008140
c6: 2494800
Y0: -2774524181546352716484898272284905686
Y1: 85331174827726081541743
# MurphyF (Bf=6.872e+10,Bg=3.436e+10,area=1.766e+18) = 2.433e-09

I'll continue searching if testsieving indicates that these are at least competitive to the snfs poly.

[swellman]

Regardless of which way 2,1109+ ultimately goes (i.e. SNFS or GNFS), do we still want to first run ECM on it as part of the “Gang of 31”? Which likely won’t start for another 4-6 weeks but I can make sure it’s first in queue.

Or are the chances of an ECM hit too low to bother delaying NFS for this particular composite by weeks or months?

Looking further ahead, if 2,1097+ proves to be within the capabilities of NFS@Home then perhaps the few composites of similar difficulty on the 1987 Cunningham base-2 list could be factored immediately while 2,1109+ undergoes ECM. Many paths forward becoming available.

[R.D. Silverman]

Originally Posted by swellman

Regardless of which way 2,1109+ ultimately goes (i.e. SNFS or GNFS), do we still want to first run ECM on it as part of the “Gang of 31”? Which likely won’t start for another 4-6 weeks but I can make sure it’s first in queue.

It is unclear to me.

Or are the chances of an ECM hit too low to bother delaying NFS for this particular composite by weeks or months?

I don't think a delay would matter. More ECM will not hurt in any case.

Looking further ahead, if 2,1097+ proves to be within the capabilities of NFS@Home then perhaps the few composites of similar difficulty on the 1987 Cunningham base-2 list could be factored immediately while 2,1109+ undergoes ECM. Many paths forward becoming available.

Whatever Greg decides is fine with me. I have no opinion. They will all get done eventually.

[frmky]

Presuming 2,1097+ goes ok, I'll do 2,2194L&M next. That'll give you some time for ECM.

[sweety439]

Originally Posted by frmky

Presuming 2,1097+ goes ok, I'll do 2,2194L&M next. That'll give you some time for ECM.

See https://homes.cerias.purdue.edu/~ssw/cun/appc722.txt, you can consider 3,715-, 2,2874M, 10,710M, 7,419- (just a suggestion)

[sweety439]

Originally Posted by charybdis

To the best of my knowledge, no number between 222 and 229 digits has ever been factored with GNFS

Are there any numbers with > 229 digits ever been factored with GNFS (a challenge, factor the 230-digit composite cofactor of R383 with GNFS)? If not, then this 221-digit cofactor of 2,2246M is the largest number which has ever been factored with GNFS

[henryzz]

Originally Posted by sweety439

See https://homes.cerias.purdue.edu/~ssw/cun/appc722.txt, you can consider 3,715-, 2,2874M, 10,710M, 7,419- (just a suggestion)

I believe he is well aware of this list.

Originally Posted by sweety439

Are there any numbers with > 229 digits ever been factored with GNFS (a challenge, factor the 230-digit composite cofactor of R383 with GNFS)? If not, then this 221-digit cofactor of 2,2246M is the largest number which has ever been factored with GNFS

A very quick google suggested that RSA 768(232 digits), RSA-240 and RSA-250 have all been factored with more digits than this.

[swellman]

Originally Posted by frmky

Presuming 2,1097+ goes ok, I'll do 2,2194L&M next. That'll give you some time for ECM.

Ok I will enqueue 2,1109+ this week. Hoping it gets processed by February, possibly sooner if Yoyo’s recent surge keeps up.

[sweety439]

Originally Posted by charybdis

2,2246M is done, meaning that 2,1109+ c225 is now the only clear GNFS target left among the remaining numbers from the 1987 edition of the Cunningham book. Let's find a good polynomial for NFS@Home to sieve with.

2^1109+1 should use SNFS instead of GNFS, see this page