20221208, 05:39  #1 
Dec 2022
D1_{16} Posts 
P1 on Fermat numbers
I'm sure it's been thought of before, but there should be some way to find P1 that's been done on Fermat numbers. If people could see the bounds previously reached, they'd be more willing to start their own runs. I, who don't think my computers could surpass what's already been done but am not sure, think this way. In any case P1 can't normally be subdivided, so should be done with the best computational resources.
TF quickly reaches diminishing returns, as is wellknown, and no new ECM Fermat factors have been found since 2011 even though recorded effort continues. So P1 looks attractive, especially for F2529 (24 has no known factor so has probably had significant outside work), but every one through 33 (which Ernst Mayer is supposed to get) should have a nonnegligible shot. Since there are so few Fermat numbers, anyone interested in high P1 would probably best contribute there if possible rather than to Mersennes. Perhaps mersenne.ca could add pages for Fermats through 33. 
20221208, 06:15  #2 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
3^{2}·1,117 Posts 
See section 1 here  https://www.mersenne.org/report_ecm/
P1 likely will not be productive with B1 less than 10 times that of ecm limits that are already performed. I vaguely remember that there should be some standalone thread about Fermat P1 deep in the forum: https://www.mersenneforum.org/showthread.php?t=15614 https://www.mersenneforum.org/showthread.php?t=20011 https://www.mersenneforum.org/showth...675#post288675 ... https://www.google.com/search?q=P1+...senneforum.org 
20221208, 13:49  #3 
Dec 2022
11·19 Posts 
None of those links are particularly useful. I have heard the '10 times' rule but really that only is an estimate for the P1 that will take the same time as _one_ ECM curve. Surely you want to run longer than that; judging by what sensible people actually do, 100 times is a better minimum.
Actually, one can directly calculate the tvalue divided by the exponent and compare that to the bounds you're thinking of using. A B1 equal to the fourth root of that gives a reasonable chance of finding a factor, and that is achievable on the larger Fermats (22 and higher), B1 equal to the cube root is good, while B1 less than the fifth root is not worthwhile. (On M1277, the worst case I know for P1, B1 would need to be near 10^15.) Ernst was going to use B1 = 10^7 on F33, and that should double with each lower exponent  well, relax it a bit on numbers with known factors, but you're still talking thousands of 'real' GHzdays. Kriesel has implied, though, that for P1 of such length, ECC memory is a must because the chance of error is too high and P1 has no real checking (maybe that's the next goal for P95 ...), which would be unfortunate because few people have ECC and you can't add it to existing systems. Anyway the main point was about some record of previous P1 attempts, and that that would be as useful as for Mersennes, and more so than for ECM. 
20221208, 15:03  #4 
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2·29·127 Posts 
ECC does great things for computation reliability. (I have a 3YEARlongandclimbing test run in LL without Jacobi check, whose interim res64s match a check run on much faster hardware. Several old ECC equipped systems have NO bad final LL residues logged in over 4 years of running each system.)
ECC can be had economically in multiple ways;

20221209, 07:01  #5 
"Mihai Preda"
Apr 2015
2^{2}×19^{2} Posts 
Concerning the splitting of P1, it is only the firststage that can't be split. So get a powerful machine to run firststage to a high B1, and afterwards fan out chunks of secondstage to multiple users.
So we're looking at a giant first stage. Well, it can be error checked! maybe the error check does need to be implemented in mprime, but it's not very hard  it's similar to the existing GEC, just a bit more complex; and just as efficient overheadwise. 
20221210, 04:24  #6 
Dec 2022
11×19 Posts 
OK, what exactly do you have in mind? Could the second stage then be done without error checking? It would require the full amount of memory on each machine (by the way, this is why we can't factor RSA1024  if the memory as well as CPU cycles could be perfectly split, it would be a feasible distributed computation).
And I'm pretty sure that method of splitting up won't work with the new 30.8/GMPECM continuation, which I'd judge necessary for F2226, so this would be limited to 2732  you can throw in MM31, it hasn't been done but there should be interest  and B2 = 100*B1 presumably. 
20221210, 13:53  #7  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2·29·127 Posts 
Quote:
Per http://www.doublemersennes.org/mm31.php MM31 has already been TF up to k=450E15, f=2kp+1 = 2 * 450E15 * (2^311) + 1 ~ 90.64 bits. MMFF can go to 96 bits on that per https://mersenneforum.org/showpost.p...&postcount=382 Last fiddled with by kriesel on 20221210 at 14:16 

20221210, 14:25  #8 
Dec 2022
11×19 Posts 
My last post was mainly intended to indicate that I would be willing to contribute cycles to such a project, if that were possible. I actually did not start this thread at all to ask or demand that anyone do such P1  merely to get information about what, if any, had been done. Obviously I wouldn't be disappointed to see it done, however.
Most people in the project use the hardware they already have, rather than buy or rent anything specifically for it; posters here may not be typical. This makes sense for a volunteer project  is not the main point simply to use the idle computing power you already have for something rather than nothing? This explains the scarcity of ECC systems, even if they aren't terribly hard to acquire. I used the calculator to find that, given the TF and ECM done, the longest runs (doubling back from B1=10M for F33, est. 20,000 GHzday stage 1, and B2 = 100*B1) would have a factor probability >5% for each of F2732 and MM31. I do know these all have known factors already. Last fiddled with by Andrew Usher on 20221210 at 14:28 
20221210, 15:55  #9  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
1CC6_{16} Posts 
Quote:
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I may add MM31 to the list of P1 reliability test exponents at https://www.mersenneforum.org/showpo...8&postcount=31 And note that used old ECC systems have been available on Ebay at times for under $250/unit (albeit only 12GB ram as listed). Last fiddled with by kriesel on 20221210 at 16:47 

20221210, 16:45  #10 
Dec 2022
11×19 Posts 
You mangled your last attempted quote.
You should know about this kind of P1 as the runs you're doing on your OBD numbers are of similar size and seem to be proceeding successfully, and also that it's preferable to further TF at the level that's been achieved. The early aborts are obviously a fixable problem (best, don't do the GCD after stage 1). I have not found any threads on actual P1 done (or attempted) on Fermat numbers except one very old one where I think the person wrote his own program as he had been unable to do the GCD (he ran stage 1 on F31 to 300,000  no factor). My 'plan' was a reply to Preda (name?) who proposed splitting the stages, which sounded good to me, and I don't think he particularly had in mind using a GPU for either. 
20221210, 16:54  #11  
"TF79LL86GIMPS96gpu17"
Mar 2017
US midwest
2×29×127 Posts 
fixed, thanks.
Quote:
Ernst is making a P1 attempt on F33 with Mlucas 20.1.1 and its stage 2 is split to multiple users IiRC. My OBD P1 attempts and M2253M in Mlucas 20.1.1 have completed stage 1 and GCD, and begun stage 2. Possibly the OBDs will be S2 split, later. Small bounds can be done quickly (days). Suitable bounds are months or years. Hence reliability is a must, and ECC helps that considerably. Last fiddled with by kriesel on 20221210 at 17:03 

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