Odd perfect related road blocks
 

Hi everyone,

Let N = p^a . M^2b (1)

be a hypothetical odd perfect number, where M is a square free positive integer and p is a special prime. McDaniel & Hagis conjectured that (1) cannot be perfect in 1975. Very recently, Yamada obtained

w(N) <= 4b^2 + 2b + 3.

Currently, I am at the final stage of working on a paper, “On odd perfect numbers of the restriction form”. In particular, I extended several authors’ early results and proved

Theorem 3. If 2b+1 < 307, then (1) is not perfect except possibly for 2b+1 = 223 and 263, the status are unknown.

The following roadblocks I am unable to handle by myself:

(223^223-1)/222 = 409989521094963541 x c504 (799)*
(409989521094963541^223-1)/409989521094963540 = 223 x c3908 (91)*

(263^263-1)/262 = c635 (390)*

* The number of ECM curves had been done on a slow p4 machine.

I need just one new factor for each case to get start. I have a strong lemma behind these computations. The success rate are very high. Please help to run a few more curves before I give it up.

Thank you in advance.

Joseph

running 400@1e7 on (223^223-1)/222 - 3GHz core2quad, should be done in 24hrs

Joseph, what B1 did you run those curves at? 799 curves at 2e3 and 799 curves at 85e7 are very different.

10metreh,

B1=1000000; B2=100000000 thought out.

[code]
GMP-ECM 6.2.2 [powered by GMP 4.2.2] [ECM]
Input number is 516248411741053652960395436827184748862511084897217762377502456224434034905125137924170344817935259950122336045143611857718767322907901227815716686379048993059061293880332637345541505919991461453890752564114634126344482510018429339411453404018945223135514150808361721533790555293536168054669954400326445726459397674031089151949275304670267658219519452338937148157536845696413116321597300233800755589805409153772759906938559062238872755146500459248479273434482651553560892669682916122705541547341688298333 (504 digits)
Using B1=11000000, B2=35133391030, polynomial Dickson(12), sigma=4043036099
Step 1 took 306607ms
Step 2 took 85165ms
********** Factor found in step 2: 732015431059055717751235492242377
Found probable prime factor of 33 digits: 732015431059055717751235492242377
Probable prime cofactor 705242526095607743265080015446043565288921781655047840484558284291328531564590341367637762791851256618356201769489720787204653401286733113504973069246617786597912347415545251135754017216052618181483502140642004356227491391452122658079363329762199969689314612990739804247835329180329247391277716676467136053161808192042000194572041912529264878194985639190553535787689788524512585864887726978565143626021507789843474186488361271520382663964712249256974106542888249972522229 has 471 digits
[/code]

Edit: Joseph, does this factorization eliminate the need for a factor of the c3908?

Alex

[code]
GMP-ECM 6.2.2 [powered by GMP 4.2.2] [ECM]
Input number is 10716687368549018763740766506156506210258720260361522976235281809991866675453511641941877971732815612425931418096560363916650395026494923817878808981644077124831733080607830496084459730811248459458809938061278635828036996709266949994249238660612006442892763114415523986136910959825342131482035732214847552279872032608309088871277760116183140576076405209606427639522415327860343074193151553224581206865415499279836545612188527758163390670461804030245602097217998369545727008825559419609534995028936152073821515552054373971674463659739536023484640994632430989874744157022140915107522171188810384994208526721910844176965299080779105345833 (635 digits)
Using B1=11000000, B2=30114149530, polynomial Dickson(12), sigma=676488604
Step 1 took 434099ms
Step 2 took 102078ms
Run 2 out of 10:
Using B1=11000000, B2=30114149530, polynomial Dickson(12), sigma=3642451195
Step 1 took 435359ms
Step 2 took 101815ms
********** Factor found in step 2: 3093592597970782253540981763792599633
Found probable prime factor of 37 digits: 3093592597970782253540981763792599633
Composite cofactor 3464156002822913916063660191851471370127924830180980072575319609465988064813804938061767992166652802325198608123611745155961838583846626169879413691616138488070021910893441777559812804188117618614976035473361505284305848039144209462534567391381069030417263236555017059390036151051159299619744603965192875096582495854908362416008344062873833282528070708156069981956230502781145912421297734891349518307027267176149202405445938495032461016522135616055774232266050676680601573518083809447182067462176727957674008757780774275442811880728643475987020419319376534912042270667918555186435478946766938221401 has 598 digits
[/code]

I guess this is my lucky day.

Alex

[quote=akruppa;171188]I guess this is my lucky day.

Alex[/quote]

Seems so! Wasn't that factor on the second curve? You are lucky :smile:

I ran on 10 cpus, so the factor of the c504 was found among the first 10 curves, the factor of the c635 among the first 20.

Alex

[QUOTE=akruppa;171188][code]
GMP-ECM 6.2.2 [powered by GMP 4.2.2] [ECM]
Input number is 10716687368549018763740766506156506210258720260361522976235281809991866675453511641941877971732815612425931418096560363916650395026494923817878808981644077124831733080607830496084459730811248459458809938061278635828036996709266949994249238660612006442892763114415523986136910959825342131482035732214847552279872032608309088871277760116183140576076405209606427639522415327860343074193151553224581206865415499279836545612188527758163390670461804030245602097217998369545727008825559419609534995028936152073821515552054373971674463659739536023484640994632430989874744157022140915107522171188810384994208526721910844176965299080779105345833 (635 digits)
Using B1=11000000, B2=30114149530, polynomial Dickson(12), sigma=676488604
Step 1 took 434099ms
Step 2 took 102078ms
Run 2 out of 10:
Using B1=11000000, B2=30114149530, polynomial Dickson(12), sigma=3642451195
Step 1 took 435359ms
Step 2 took 101815ms
********** Factor found in step 2: 3093592597970782253540981763792599633
Found probable prime factor of 37 digits: 3093592597970782253540981763792599633
Composite cofactor 3464156002822913916063660191851471370127924830180980072575319609465988064813804938061767992166652802325198608123611745155961838583846626169879413691616138488070021910893441777559812804188117618614976035473361505284305848039144209462534567391381069030417263236555017059390036151051159299619744603965192875096582495854908362416008344062873833282528070708156069981956230502781145912421297734891349518307027267176149202405445938495032461016522135616055774232266050676680601573518083809447182067462176727957674008757780774275442811880728643475987020419319376534912042270667918555186435478946766938221401 has 598 digits
[/code]

I guess this is my lucky day.

Alex[/QUOTE]

Can you quickly run a dozen of curves on M1061 or F14? :wink:

[QUOTE=akruppa;171195]I ran on 10 cpus, so the factor of the c504 was found among the first 10 curves, the factor of the c635 among the first 20.[/QUOTE]

The real question is, how come your are using an old version of your own software with an even older version of GMP? :whistle: :razz:

Those machines are a bit picky about which binaries they like, iirc I had to link some libraries statically and some others dynamically to get GMP-ECM to run at all, so I wasn't too keen on updating the binaries. But now that GMP 4.3.0 is out, it's worthwhile to do it.

Alex