Polynomial selection for 2,1109+ c225
2,2246M is [URL="https://mersenneforum.org/showthread.php?t=27356"]done[/URL], 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.
The composite is [code]126451876805119252959661548967232013601866431183534308908427174011662073024932261126233275630388431500884665768427172907593022873289431612806891303891687570778305960728323476541491713906981621716831593952842244282430058291761[/code] To the best of my knowledge, no number between 222 and 229 digits has ever been factored with GNFS, so there are no nearby record scores to compare with, but extrapolating from other degree 6 scores suggests we should aim for a score of at least 2e16. For CADO users, something like P=15M, incr=420, nq=46656, sopteffort=10, ropteffort=100 would be a good start parameterwise (and don't forget degree=6). Those with good GPUs may wish to try msievegpu stage 1 followed by CADO sopt, which is a technique that has produced some exceptional polynomials in the past; if you would like to try this, ensure that you have either the latest CADO or a version from before 11 March this year, otherwise you will encounter [URL="https://mersenneforum.org/showthread.php?p=612689#post612689"]this bug[/URL]. There's no rush, don't worry if you need to wait for cooler weather. I intend to start searching with CADO in a week or so. 
Just to get the party started, here are a couple of early hits.
[CODE]R0: 890858233525730603070019268730360731 R1: 10637678344873644797 A0: 299836229189041894895839186985662895951560076832 A1: 237194224808389178293161225128621163060026 A2: 46081070789360664784376386970027831 A3: 51529985200329980370627087588 A4: 18026757896690121716396 A5: 2852764153396262 A6: 252972720 skew 4096052.87, size 1.995e16, alpha 8.045, combined = 1.486e16 rroots = 4 R0: 1543011722806615565935575378314030175 R1: 842031878738877989 A0: 6350450293847067402197405075596797572174622865528576 A1: 3139610947120576605897442761369137813674879640 A2: 843606577109644671310292753045506226718 A3: 6622510117319190858833637755861 A4: 734856321473040043162292 A5: 2244560413626756 A6: 9369360 skew 41379413.60, size 1.738e16, alpha 12.221, combined = 1.288e16 rroots = 4[/CODE] 
Here's a spin for the previous two:[code]Y0: 890858233519512848164405588725579840
Y1: 10637678344873644797 c0: 20679080292568510209376599202651938080731751 c1: 16905369045270557188781384384089427958 c2: 2071016516892570126261873688125653 c3: 647249845227500980682508056 c4: 25067605669229211438126 c5: 1965584270847302 c6: 252972720 skew: 313566.209 # lognorm 60.98, E 53.93, alpha 7.05 (proj 2.10), 6 real roots # MurphyE(Bf=1.000e+07,Bg=5.000e+06,area=1.000e+16)=1.983e16 Best poly cownoise values: 589597.92262 2.01167412e16[/code] 
I'll try casually. Msieve doesn't seem good. ( can't get 1.53 gpu to run due to ptx incompatibility. GTX 1660 TI and cuda 7.5 )

After a couple A40months, I found two more higher scoring polys from msieve.
[CODE]# norm 2.363604e16 alpha 9.855342 e 1.471e16 rroots 4 skew: 4188324.95 c0: 4890952461071308375125802645743946534044245681184 c1: 1464426985010460823704889708794214954589992 c2: 597300025874739851693961141580723240 c3: 434858756700091304313330515360 c4: 60480910188020917238229 c5: 36012170114448258 c6: 2327925600 Y0: 615402652607881386341306354933736125 Y1: 1564161336280197503 # norm 2.516901e16 alpha 11.340645 e 1.540e16 rroots 4 skew: 5962737.46 c0: 36610384221328653794084765331280784334500166002560 c1: 6597385732367497117864187808228061224463212 c2: 3597302716498354719019930592716700096 c3: 998777198088521283283165788933 c4: 364143521396930855524484 c5: 52296633709802010 c6: 3398194800 Y0: 577802612761468877350169963836667229 Y1: 42417278640103023257 [/CODE] But using Ed's script, they didn't improve quite as much as the early hit above. [CODE]Y0: 615402652608941593664973108163639561 Y1: 1564161336280197503 c0: 99837092116108261904550451646795621673004800 c1: 2024046815793900142308556853273336977440 c2: 15463527646675689570225765198286064 c3: 119928244381367488132162484048 c4: 166485563730524246399709 c5: 26544794673725058 c6: 2327925600 skew: 467990.840 # lognorm 63.90, E 54.90, alpha 9.00 (proj 2.97), 6 real roots # MurphyE(Bf=1.000e+07,Bg=5.000e+06,area=1.000e+16)=1.990e16 Best poly cownoise values: 771030.94076 1.99451279e16 [/CODE] 
My first run with CADO:
[code]c0: 2310835730251156327778236511197625826302780208 c1: 442704803492886905636371260028570831748 c2: 32698967345475004746748591472162564 c3: 4077124657453197102154489355 c4: 35221438603837936231384 c5: 21786327094255250 c6: 5319514200 Y0: 859967642928493740637122025458452125 Y1: 188619699713867324748872261 # MurphyE (Bf=6.872e+10,Bg=3.436e+10,area=1.766e+18) = 2.033e09[/code] Cownoise stats: skew 1025513.35579 score 1.62798458e16 This was admin 3e8, admax 35e7, incr 60060. I'll continue searching up way high to stay out of the way of the regular searchers. 
Let's not forget that we also have a free polynomial with
[CODE]R0: 49039857307708443467467104868809893875799651909875269632 R1: 1 A0: 2 A1: 0 A2: 0 A3: 0 A4: 0 A5: 0 A6: 1 skew 1.12, size 2.613e16, alpha 1.888, combined = [B]1.760e16[/B] rroots = 0 [/CODE] We need to beat that significantly. 
[QUOTE=Batalov;613248]Let's not forget that we also have a free polynomial with
[CODE]R0: 49039857307708443467467104868809893875799651909875269632 R1: 1 A0: 2 A1: 0 A2: 0 A3: 0 A4: 0 A5: 0 A6: 1 skew 1.12, size 2.613e16, alpha 1.888, combined = [B]1.760e16[/B] rroots = 0 [/CODE] We need to beat that significantly.[/QUOTE] I expected that it wouldn't be too hard to beat that escore, and I was right. What I hadn't counted on was the SNFS poly overperforming its escore by over 40%(!) compared to the two topscoring GNFS polys. Maybe SNFS is faster after all; we probably need a GNFS poly scoring ~2.5e16 to compete with it. I think we've reached a point where the parameters used to generate the msieve/cownoise escores are so far from what will actually be used during sieving that the scores only bear a vague resemblance to sieving speed. Plugging some more realistic values into CADO's polyselect3 (Bf 34359738368.0 Bg 34359738368.0 area 2147483648000000000.0) gives the SNFS poly a score almost twice as high as the GNFS polys. 
I went very granular and got
[code] skew: 685194.469 c0: 13546832639784934995088934891473021436914757595 c1: 243239753933587411699854385797238306940956 c2: 18952157534337102628447505483356966 c3: 653316556750163296492944500212 c4: 2574375769149943375426741 c5: 195684033327748056 c6: 81908173620 Y0: 1041043757015113400466899425916289274 Y1: 2830716526317416698316524373 # MurphyE (Bf=6.872e+10,Bg=3.436e+10,area=1.766e+18) = 1.756e09 [/code] 
second run
[code] n: 126451876805119252959661548967232013601866431183534308908427174011662073024932261126233275630388431500884665768427172907593022873289431612806891303891687570778305960728323476541491713906981621716831593952842244282430058291761 skew: 697478.803 c0: 26487282380138522677004084831483303431066050150 c1: 24753607872970529743670167294611358561023 c2: 432036146231461013035116862579046871 c3: 257470942642010751446884931927 c4: 1341532846594386074140153 c5: 223518134738310192 c6: 233429676480 Y0: 1033071967615446320814534193404610478 Y1: 21370116632131503382859723 # MurphyE (Bf=6.872e+10,Bg=3.436e+10,area=1.766e+18) = 1.776e09 [/code] 
firejuggler and I are using the same CADO sieve/Murphyscore params (taken from the c230.params file) for score evaluation.
Here's my secondrun best: [code]skew: 349620.103 c0: 35261944489889081487427593499134306423434235 c1: 1147840847825503997219617398569984615228 c2: 6032823929352871049090059774838128 c3: 8280172716447232050639984506 c4: 93116889162997251267859 c5: 46149285013390062 c6: 9179450280 Y0: 813240839151117999538946984841129674 Y1: 4775593547116029134541157 # MurphyE (Bf=6.872e+10,Bg=3.436e+10,area=1.766e+18) = 2.210e09[/code] I've edited the post from my first poly to show this CADO score also. This one is 9% better, from admin 35e7 admax 45e7 incr 60060. It seems to me that scoring 2e16 on cownoise is fairly easy. A 2.2 should be reachable without difficulty, but getting to 2.5 to beat the SNFS poly will take some luck or lots of work. We need the combo of a poly scoring 2.0+ before spin plus a lucky extra20% spin. In CADOscore terms, a 2.4 before spin with a lucky spin is likely necessary to beat the SNFS poly; also, we can't be sure that "spinning" to optimize regular Escore actually sieves better than these CADO polys that score best on a sieve region we're actually going to use. That is, "spin" may be a bit of an illusion on deg 6 sized jobs. I'm willing to keep searching, but I'll pause until we decide whether to just use the SNFS poly for the job. For those of you searching with CADO, I suggest you use the params.c230 file as your template, and edit the poly select parameters to suit your run. Doing so will produce CADO Murphy scores consistent with those on this thread, reducing our labor to check cownoise and improving the predictive power of the score. 
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