20090427, 14:59  #1 
May 2005
2^{2}·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: (223^2231)/222 = 409989521094963541 x c504 (799)* (409989521094963541^2231)/409989521094963540 = 223 x c3908 (91)* (263^2631)/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 
20090427, 15:31  #2 
(loop (#_fork))
Feb 2006
Cambridge, England
18EF_{16} Posts 
running 400@1e7 on (223^2231)/222  3GHz core2quad, should be done in 24hrs

20090427, 17:07  #3 
Nov 2008
912_{16} 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 20090427 at 17:09 
20090427, 17:29  #4 
May 2005
2^{2}·3·5 Posts 
10metreh,
B1=1000000; B2=100000000 thought out. 
20090427, 18:05  #5 
"Nancy"
Aug 2002
Alexandria
2467_{10} Posts 
Code:
GMPECM 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 Alex Last fiddled with by akruppa on 20090427 at 18:44 
20090427, 18:32  #6 
"Nancy"
Aug 2002
Alexandria
2,467 Posts 
Code:
GMPECM 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 Alex 
20090427, 18:56  #7 
Nov 2008
2·3^{3}·43 Posts 
Seems so! Wasn't that factor on the second curve? You are lucky
Last fiddled with by 10metreh on 20090427 at 18:56 
20090427, 19:00  #8 
"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 
20090427, 19:08  #9  
Oct 2004
Austria
2·17·73 Posts 
Quote:
Last fiddled with by Andi47 on 20090427 at 19:09 

20090427, 20:03  #10 
Jun 2003
Ottawa, Canada
2^{2}×293 Posts 

20090427, 20:11  #11 
"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 GMPECM 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 
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