Odd perfect related road blocks

jchein1 · 2009-04-27, 14:59

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

fivemack · 2009-04-27, 15:31

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

10metreh · 2009-04-27, 17:07

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

jchein1 · 2009-04-27, 17:29

10metreh,

B1=1000000; B2=100000000 thought out.

akruppa · 2009-04-27, 18:05

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

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

Alex

akruppa · 2009-04-27, 18:32

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

I guess this is my lucky day.

Alex

10metreh · 2009-04-27, 18:56

Quote:

Originally Posted by akruppa

I guess this is my lucky day.

Alex

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

akruppa · 2009-04-27, 19:00

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

Andi47 · 2009-04-27, 19:08

Quote:

Originally Posted by akruppa

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

I guess this is my lucky day.

Alex

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

Jeff Gilchrist · 2009-04-27, 20:03

Quote:

Originally Posted by akruppa

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.

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

akruppa · 2009-04-27, 20:11

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

2009-04-27, 14:59	#1
jchein1 May 2005 2²·3·5 Posts	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

2009-04-27, 17:07	#3
10metreh Nov 2008 2·3³·43 Posts	Joseph, what B1 did you run those curves at? 799 curves at 2e3 and 799 curves at 85e7 are very different. Last fiddled with by 10metreh on 2009-04-27 at 17:09

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2009-04-27, 15:31	#2
fivemack (loop (#_fork)) Feb 2006 Cambridge, England 7²×131 Posts	running 400@1e7 on (223^223-1)/222 - 3GHz core2quad, should be done in 24hrs

2009-04-27, 17:29	#4
jchein1 May 2005 2²×3×5 Posts	10metreh, B1=1000000; B2=100000000 thought out.

2009-04-27, 19:00	#8
akruppa "Nancy" Aug 2002 Alexandria 2,467 Posts	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

2009-04-27, 20:11	#11
akruppa "Nancy" Aug 2002 Alexandria 2,467 Posts	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