Right Perfect Prime Numbers - Page 2

JeppeSN · 2016-01-23, 16:09

Quote:

Originally Posted by JeppeSN

Can we extend this?

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

/JeppeSN

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

ATH · 2016-01-23, 17:00

Yeah for 57885161 there is also a factor: 22127627.

For p=74207281, I trial factored 2^p-1*(2^p-1) + 1 up to 4.46*10¹² with no factor.

JeppeSN · 2016-01-26, 12:43

Quote:

Originally Posted by ATH

For p=74207281, I trial factored 2^p-1*(2^p-1) + 1 up to 4.46*10¹² with no factor.

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

ATH · 2016-01-26, 17:20

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.

JeppeSN · 2016-01-26, 20:34

As R. Gerbicz said in his last post to this thread, an n-1 test should be possible. /JeppeSN

ATH · 2016-01-27, 06:38

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.

Batalov · 2016-01-27, 08:43

Quote:

Originally Posted by ATH

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.

PFGW can. The limit is the same (gwnum library is the same) - FFT max size of 32M which is more than enough, but...

Quote:

Originally Posted by ATH

A 44 million digit PRP LL test would take ~~[COLOR=DarkRed]10-14[/COLOR]~~ 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.

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).

henryzz · 2016-01-27, 12:36

Quote:

Originally Posted by Batalov

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).

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.

JeppeSN · 2016-01-27, 14:04

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"

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: http://sourceforge.net/projects/openpfgw/

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

/JeppeSN

paulunderwood · 2016-01-27, 15:40

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

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

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

JeppeSN · 2016-01-27, 16:52

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

2016-01-27, 14:04	#20
JeppeSN "Jeppe" Jan 2016 Denmark 2×83 Posts	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" 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: http://sourceforge.net/projects/openpfgw/ Can anyone else get to a point where the N-1 test actually progresses? /JeppeSN

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2016-01-23, 17:00	#13
ATH Einyen Dec 2003 Denmark 101111010111₂ Posts	Yeah for 57885161 there is also a factor: 22127627. For p=74207281, I trial factored 2^p-1(2^p-1) + 1 up to 4.4610¹² with no factor.

2016-01-26, 17:20	#15
ATH Einyen Dec 2003 Denmark 7·433 Posts	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.

2016-01-26, 20:34	#16
JeppeSN "Jeppe" Jan 2016 Denmark 2·83 Posts	As R. Gerbicz said in his last post to this thread, an n-1 test should be possible. /JeppeSN

2016-01-27, 06:38	#17
ATH Einyen Dec 2003 Denmark 7×433 Posts	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.

2016-01-27, 15:40	#21
paulunderwood Sep 2002 Database er0rr 111000000110₂ Posts	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 Here, I would run on linux: ./pfgw64 -q"2^74207280*(2^74207281-1)+1" -f0 The chance it will be prime is next to zero, but you will know!

2016-01-27, 16:52	#22
JeppeSN "Jeppe" Jan 2016 Denmark 166₁₀ Posts	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