Status of Wagstaff testing? and testing Mersenne primes for Wagstaff-ness

[axn]

31000-33000 range

2^31019+1 = 36606080243
2^31033+1 = 37602780478825465675483
2^31033+1 = 5274911256999257402385523
2^31039+1 = 15107763782145257
2^31039+1 = 2696645101027739
2^31063+1 = 204659511295533506935963
2^31063+1 = 2459021172784025850457123
2^31079+1 = 1277922561229512977
2^31079+1 = 742254725505350547571
2^31091+1 = 5523841497517850569483
2^31091+1 = 8271765629721199361602481
2^31121+1 = 18850362981826478419075193
2^31147+1 = 6506780792059484986593698654281
2^31159+1 = 119724070256997970081
2^31159+1 = 2802563561650167604097
2^31177+1 = 11213724158959716019
2^31181+1 = 40712330002777
2^31183+1 = 118503930982958697195644551123
2^31189+1 = 14238768043105024609
2^31193+1 = 255360621097
2^31193+1 = 42530990255995354572275041
2^31249+1 = 3719124991924565051
2^31267+1 = 110134308433177831201728491
2^31277+1 = 1025970948126409451020341307147
2^31327+1 = 218594321552031025485547414177
2^31333+1 = 6202479086153575618763728459
2^31337+1 = 365015928519211619228377688833
2^31379+1 = 356411154331
2^31379+1 = 52055187923
2^31391+1 = 112913352265210696231797707
2^31393+1 = 17145060851398187
2^31469+1 = 1951459697067577
2^31469+1 = 3502361297421221333628210179
2^31477+1 = 60982260792796794657906737
2^31481+1 = 28830003790332534145507
2^31511+1 = 7953595075739102233084224505939
2^31517+1 = 153688062752403823286814121
2^31517+1 = 4930182170482960049089
2^31543+1 = 1254151095738854347
2^31543+1 = 15155197743355596931
2^31543+1 = 188337822998723
2^31547+1 = 1370919258238605165319987
2^31547+1 = 1550345168289696740737
2^31573+1 = 114782624849401
2^31573+1 = 1955500388575970267
2^31573+1 = 29487692827883
2^31601+1 = 495086555885477107
2^31607+1 = 12833547475243123
2^31607+1 = 39240620443957139393
2^31627+1 = 2384449060397716139
2^31627+1 = 360683229087339699433787
2^31627+1 = 99391080005849321
2^31643+1 = 44189041368587
2^31657+1 = 1036411181804874580356011
2^31663+1 = 8663244266350500480689660953
2^31667+1 = 3209528050261109988739649
2^31699+1 = 156842957861939
2^31723+1 = 23466170429308351842203
2^31751+1 = 24948086377594811
2^31751+1 = 4206392486368603
2^31751+1 = 90223260291763171
2^31793+1 = 29528886583076364977881
2^31793+1 = 56857512169093784593
2^31793+1 = 7100412843113
2^31799+1 = 201448575574994625529
2^31817+1 = 1106540888565041
2^31817+1 = 94915516548260491
2^31849+1 = 2845507044378448897
2^31849+1 = 79955342328194603
2^31859+1 = 21763215711715891169
2^31859+1 = 25673831025461903513
2^31873+1 = 12953215293009321752003
2^31873+1 = 89571808088547067
2^31883+1 = 1953905018801
2^31907+1 = 56281616712939139
2^31957+1 = 280833302371500996558064892449
2^31957+1 = 375021950276918773511587
2^31963+1 = 202701397797036043
2^31991+1 = 2655881239259
2^32003+1 = 4034839352712329827
2^32009+1 = 154246394640742170242807896769
2^32029+1 = 30707767118192557221069657001167817
2^32051+1 = 41364914183256811222073
2^32059+1 = 227222073190668569732452729
2^32063+1 = 14221872415938449083394092649
2^32063+1 = 1476869233241
2^32063+1 = 481295267241673
2^32069+1 = 16475196256117206740129
2^32069+1 = 821590968233530440302654346499
2^32083+1 = 5313686538537415615788587
2^32089+1 = 770180253105587
2^32117+1 = 8133839640314371
2^32119+1 = 15762867089524942521878987
2^32119+1 = 770964970991768011
2^32143+1 = 10538414766868369211445177836033
2^32143+1 = 488834835869874234117734912089
2^32183+1 = 22907599933109273385349761353
2^32191+1 = 115988035921
2^32213+1 = 17481288740841227103653993
2^32213+1 = 19864303198459
2^32213+1 = 7004020533793
2^32251+1 = 102461950405952709608662537
2^32251+1 = 1082887823235344810725281209
2^32251+1 = 194480155089308626159009
2^32261+1 = 20035167004943108272937
2^32261+1 = 284892291636291439273755734153
2^32261+1 = 8960922927525367027
2^32299+1 = 6859995781488417925241
2^32299+1 = 9714711196579667942789897
2^32309+1 = 528992441125194883
2^32321+1 = 110314967258090439165449
2^32321+1 = 19548881032669556933777
2^32323+1 = 112017905448174153386953
2^32323+1 = 150484261921914553
2^32323+1 = 374534983361537203
2^32377+1 = 1096170423492101317447441
2^32381+1 = 229818844077098285290822753
2^32401+1 = 12283015982652105937
2^32401+1 = 313709461855169
2^32411+1 = 8469881767870256579
2^32441+1 = 218976532166668822232441
2^32479+1 = 102594340411
2^32479+1 = 161911654654323443
2^32479+1 = 48951235249170377
2^32479+1 = 9048734097232340508716258723
2^32503+1 = 332079073645531837073
2^32537+1 = 35512508651
2^32537+1 = 515890733620403
2^32563+1 = 48211940575854669833
2^32569+1 = 2686336096486241
2^32569+1 = 536653053678857
2^32573+1 = 5967144942278589055954043
2^32579+1 = 74346629120914219
2^32603+1 = 191269307779486921
2^32603+1 = 9313536190619
2^32621+1 = 6617638808116327427
2^32621+1 = 80841164087689619866276265499329
2^32633+1 = 20704123369716131
2^32633+1 = 66291350494344440833
2^32633+1 = 83176245922043
2^32647+1 = 2490963022245480733043
2^32687+1 = 2312863724271662299
2^32707+1 = 175670507562950044616608657
2^32713+1 = 1688940684727378005212763528403
2^32717+1 = 15140183610295019483
2^32717+1 = 73712912581215203
2^32749+1 = 127806462751729929976988537
2^32749+1 = 822763844266061653642875779
2^32789+1 = 2241035428544443
2^32789+1 = 284616454939
2^32789+1 = 48393678569
2^32797+1 = 441006813487097706920689
2^32801+1 = 1774839018097
2^32803+1 = 201960985553
2^32831+1 = 48839511362107
2^32833+1 = 21586533441429999980159153
2^32839+1 = 202873512709703201
2^32843+1 = 158257194106457433569
2^32843+1 = 33700468092622182014430484249
2^32843+1 = 9096613171504318617600431761
2^32869+1 = 311191050788338180723
2^32909+1 = 4555345024778819801874439209979
2^32909+1 = 866791730633
2^32911+1 = 589105255651317323
2^32911+1 = 60251236321455100763539
2^32917+1 = 69732228612833771015441
2^32933+1 = 429789704854085417867
2^32933+1 = 8269997574955395036451
2^32939+1 = 486826270513356865653860593
2^32941+1 = 255973706353
2^32941+1 = 664335677489904571609
2^32957+1 = 747655621398500993555483
2^32971+1 = 16507553749495887019
2^32983+1 = 139967167125222523
2^32987+1 = 50238411426599440816555994129
2^32987+1 = 70054165025290522051
2^32993+1 = 141076677979829475388117601369
2^32999+1 = 2244279413473
2^32999+1 = 350995249459
2^32999+1 = 818078604261655777099

[GP2]

Mersenne primes and Wagstaff primes do not share the same exponents, beyond the very small exponents that inspired the new Mersenne conjecture: 3, 5, 7, 13, 17, 19, 31, 61, 127.

But what about exponents of fully-factored composite Mersenne numbers and exponents of fully-factored composite Wagstaff numbers? I wondered if there are any correlations and compared the two lists. Links to them can be found on this page and on this page.

I ignored the range under 1200, where the Cunningham project has fully factored nearly everything. Apart from those, the two lists have the following exponents in common: 1459, 7417, 8849.

We can probably ignore the first one (1459) because it is still pretty close to the Cunningham project range of very small exponents. But I was kind of surprised to see the two larger ones (7417 and 8849) because it seems like too much of a coincidence.

In the range between 7000 and 8000, there are 107 prime numbers. None of these are exponents of Mersenne primes or Wagstaff primes. Only 5 out of 107 are exponents of fully-factored Mersenne numbers and only 1 out of 107 is an exponent of a fully-factored Wagstaff number... but 7417 is a member of both subsets.

In the range between 8000 and 9000, there are 110 prime numbers. None of these are exponents of Mersenne primes or Wagstaff primes. Only 2 out of 110 are exponents of fully-factored Mersenne numbers and only 4 out of 110 are exponents of fully-factored Wagstaff numbers... but 8849 is a member of both subsets.

The largest factor of M7417 (other than the prime cofactor) is 25 digits long. The largest factor of W7417 is 33 digits long. The largest factor of M8849 is 20 digits long. The largest factor of W8849 is 30 digits long.

Obviously the set of fully-factored exponents is not statically defined, it grows slowly over time as we factor to greater depths.

And there are no coincidences in the ranges 2k, 3k, 4k, 5k, 6k, or 9k... or higher. So far, anyway, perhaps we might find some if we do deeper ECM testing (beyond t30 for Wagstaff or beyond t40 for Mersenne).

But still, what are the odds of randomly selecting 5 objects out of 107 without replacement, and then selecting 1 object out of 107, and it turns out that the intersection of the two selections is non-empty? And then repeating the exercise with 2 out of 110 and then 4 out of 110, and again end up with a non-empty intersection?

[science_man_88]

Originally Posted by GP2

But still, what are the odds of randomly selecting 5 objects out of 107 without replacement, and then selecting 1 object out of 107, and it turns out that the intersection of the two selections is non-empty? And then repeating the exercise with 2 out of 110 and then 4 out of 110, and again end up with a non-empty intersection?

First, it depends on if the selections of 107 are the same. If so, it's obvious 5/107 ~4.67% . The second one, I'm too lazy to work out but would suspect about 1/55

[GP2]

Originally Posted by science_man_88

First, it depends on if the selections of 107 are the same. If so, it's obvious 5/107 ~4.67% . The second one, I'm too lazy to work out but would suspect about 1/55

Yeah, I don't mean astronomical odds, I just mean "makes you go hmmm" odds. And not just one, but the two cases occurring together.

There are 988 not-fully-factored exponents of Wagstaff numbers in the 0 to 10000 range. I'm going to run ECM on them to t=40, trying to create more fully-factored exponents. I'll start slowly and then throw more resources at it if anything turns up in the 2k or 3k ranges.

[GP2]

In a possibly related vein, there's a few people doing fairly deep factorization of Wagstaff numbers with Mersenne-prime exponents and then PRP testing the cofactors.

So far there are no PRPs. The page is here.

[axn]

Originally Posted by GP2

There are 988 not-fully-factored exponents of Wagstaff numbers in the 0 to 10000 range. I'm going to run ECM on them to t=40, trying to create more fully-factored exponents. I'll start slowly and then throw more resources at it if anything turns up in the 2k or 3k ranges.

Right now I'm running t35 on these (started at 1300, currently at around 2800). t40 would probably have more success. But it would probably be better to work your way down from 10500, rather than up from 1300. It does look like t40 or similar has been already run on the lower ones, so odds of success will be low there. But if you're starting from 1300 onwards, I'll stop my t35 run (after it reaches 3000).

[GP2]

Originally Posted by axn

Right now I'm running t35 on these (started at 1300, currently at around 2800). t40 would probably have more success. But it would probably be better to work your way down from 10500, rather than up from 1300. It does look like t40 or similar has been already run on the lower ones, so odds of success will be low there. But if you're starting from 1300 onwards, I'll stop my t35 run (after it reaches 3000).

OK, I only started a couple of hours ago, so I switched to doing t35, working downward from 10,000. I guess we can coordinate at some point about meeting in the middle.

I am omitting the exponents 2203, 2281, 3217, 4253, 4423, 9689, 9941, which have already been deeply ECM'd by the Mersenneplustwo project I mentioned a couple of messages ago.

(And 1279 is already fully-factored. When I mentioned that there aren't any Wagstaff PRPs for Mersenne-prime exponents, obviously I wasn't including the very small exponents.)

Have you found any new factors so far?

[axn]

Originally Posted by GP2

Have you found any new factors so far?

Just the two from post 192. Which, btw, was totally unexpected -- I would've expected all p40s and smaller to have been found already, for such small exponents. I wouldn't expect anymore success from t35 for a while - hopefully, working backwards would be more productive.

[GP2]

Originally Posted by axn

It does look like t40 or similar has been already run on the lower ones, so odds of success will be low there.

Here are the current counts of known factors of size 35 digits or more in each range:

1k        46
2k        31
3k         3
4k         5
5k         1
6k         8
7k         5
8k         3
9k         5

So yes, there definitely was more done in the 1k and 2k ranges. But even in the higher ranges I wonder if it wasn't done to t=35 already.

There are roughly 110 to 120 prime exponents in each range, with the 1k and 2k ranges at around 130.

[axn]

Originally Posted by GP2

So yes, there definitely was more done in the 1k and 2k ranges. But even in the higher ranges I wonder if it wasn't done to t=35 already.

It is likely that some sporadic t35 or higher was run on the 3k-10k ranges, so there maybe some that were left out. Even if a systematic t35 has been run already, another round of t35 will find some more factors. A full t40 will be about 8-9x as costly as a t35, so that is a substantial investment of time.

Perhaps we could switch places. You can take 3k onwards to t40 (using P95 + GMP-ECM - just P95 alone would not be most efficient). I can work my way downwards from 10k to t35. What do you think?

[DukeBG]

Factors of the 33000-34000 range were added to FactorDB by someone.