[QUOTE=JeppeSN;423710]Can we extend this?

[B]p: factor(s) of 2p-1*(2p-1) + 1[/B]
57885161: [B]7[/B]
74207281: ?

/JeppeSN[/QUOTE]

Update:

42643801: 3593, 7089208037
43112609: 7, 211, 70121, 71647, 1846524311 (was in 2008 post by ATH)
57885161: 7, 22127627
74207281: ?

My trial factoring with the latest perfect number, incremented by one, has not yielded any divisors under $75\cdot 10^9$.

/JeppeSN

Yeah for 57885161 there is also a factor: 22127627.

For p=74207281, I trial factored 2[sup]p-1[/sup]*(2[sup]p[/sup]-1) + 1 up to 4.46*10[sup]12[/sup] with no factor.

[QUOTE=ATH;423758]For p=74207281, I trial factored 2[sup]p-1[/sup]*(2[sup]p[/sup]-1) + 1 up to 4.46*10[sup]12[/sup] with no factor.[/QUOTE]

Do you know if anyone is going further with trial factoring this beast, $2^{74207280}(2^{74207281}-1)+1$, or even do a primality test of it? It may turn out to be very hard to find a prime factor for this one? /JeppeSN

Yeah the chance of finding a factor is very low, and doing a primality test is impossible. At 44.7 million digits only an LL test would be fast enough but it will not work on a number of this form. There is no software I can think of that can even do a PRP test.

Anyway these numbers are not really that important I think, it was just a question a user asked almost 8 years ago which turned into this tiny trial factoring effort. I have never seen any interest in "perfect numbers + 1" anywhere else.

As R. Gerbicz said in his last post to this thread, an [URL="http://primes.utm.edu/prove/prove3_1.html"]n-1 test[/URL] should be possible. /JeppeSN

Yes true, but that was for 79502 digits. I do not think programs like PFGW, which can do the N-1 test, can handle 44 million digits or anywhere near this size.

A 44 million digit LL test would take 10-14 days on the very fastest computers / graphic cards with Prime95 / CudaLucas, and an N-1 test would take longer than that. I do not think PFGW can save the test during it and restart, but I'm not certain about that.

[QUOTE=ATH;424260]Yes true, but that was for 79502 digits. I do not think programs like PFGW, which can do the N-1 test, can handle 44 million digits or anywhere near this size.
[/QUOTE]
PFGW can. The limit is the same (gwnum library is the same) - FFT max size of 32M which is more than enough, but...
[QUOTE=ATH;424260]A 44 million digit PRP [STRIKE]LL[/STRIKE] test would take [STRIKE][COLOR=DarkRed]10-14[/COLOR][/STRIKE] days on the very fastest computers [STRIKE]/ graphic cards[/STRIKE] with Prime95 [STRIKE]/ CudaLucas[/STRIKE], and an N-1 test would take longer than that. I do not think PFGW can save the test during it and restart, but I'm not certain about that.[/QUOTE]
Because of the form of the number, LL test has no relevance and fast special FFT is inapplicable, so you can safely triple the estimate (and much more than triple if you only use single thread with a vanilla binary). PFGW is single-thread, but you can try to rig Prime95's multithreaded code on zero-padded FFT and write a custom mod step (2^N-2^M+1 is much better than a general number).

[QUOTE=Batalov;424267]PFGW can. The limit is the same (gwnum library is the same) - FFT max size of 32M which is more than enough, but...

Because of the form of the number, LL test has no relevance and fast special FFT is inapplicable, so you can safely triple the estimate (and much more than triple if you only use single thread with a vanilla binary). PFGW is single-thread, but you can try to rig Prime95's multithreaded code on zero-padded FFT and write a custom mod step (2^N-2^M+1 is much better than a general number).[/QUOTE]
Based upon my fiddling with PFGW the FFT length would be probably be 4x as large as the mersenne prime. It would also be general reduction as well.
Possible if someone wants to spend several months on it I think.

On my Windows system, it crashes.

I try with:

[CODE]
.\pfgw64.exe -q"2^74207280*(2^74207281-1)+1" -t -f0 -h"jeppe_M74207281.txt"
[/CODE]where -t should enforce a deterministic N-1 test, -f0 should skip factoring, and the jeppe_M74207281.txt is a local help file containing just one line with 2^74207281-1 on it. The help file is for making PFGW quickly realize it has the full factorization of N-1 (our even perfect number). Apparently, PFGW does not read news sites on the web telling about the recent discovery of M74207281, so I will have to help it.

pfgw32.exe with the same parameters crashes as well, on my machine here.

Using the binaries from .7z file here: [URL]http://sourceforge.net/projects/openpfgw/[/URL]

Can anyone else get to a point where the N-1 test actually progresses?

/JeppeSN

It is much quicker to run a default 3-PRP test, rather than a fully blown N-1 test. If that goes fine, I am sure the authors will fix up PFGW for you :wink:

Here, I would run on linux: [c]./pfgw64 -q"2^74207280*(2^74207281-1)+1" -f0[/c]

The chance it will be prime is next to zero, but you will know! :smile:

Maybe it is smarter to sieve even further before attempting a PRP or primality test. After all, what we want to add to ATH's table above is the smallest prime factor, and it is too optimistic to hope the primality test will result in N being its own smallest prime factor. /JeppeSN