Mersenne number factored (disbelievers are biting elbows)

[petrw1]

As I get older I notice 2 things starting to happen:

1. I repeat myself
2. I repeat myself

[xilman]

Originally Posted by Dr Sardonicus

"It's not nice to make people spray coffee all over their monitors."

Aka C|N>K


Or have you forgotten that too?

[Dr Sardonicus]

Originally Posted by xilman

Aka C|N>K


Or have you forgotten that too?

I have never seen that before. Therefore I have not forgotten it.

[Maciej Kmieciak]

The 350th fully-factored or probably-fully-factored Mersenne number with prime exponent (not including the Mersenne primes themselves) is M1399.

The most recent factor (61 digits) was found by Ryan Propper on December 19 (UTC) and the PRP test was done by mikr and myself. There are 3 factors in all, plus the cofactor.

[Dr Sardonicus]

Originally Posted by Maciej Kmieciak

The 350th fully-factored or probably-fully-factored Mersenne number with prime exponent (not including the Mersenne primes themselves) is M1399.

The most recent factor (61 digits) was found by Ryan Propper on December 19 (UTC) and the PRP test was done by mikr and myself. There are 3 factors in all, plus the cofactor.

FWIW, I ran Pari-GP's isprime() on this PRP308 with the following result:

? n=(2^1399-1)/28875361/4320651071020341609502042221583629017824960697/9729831901051958663829453004687723271026191923786080297556081;

? isprime(n)

%2 = 1

It didn't take very long.

The manual entry says

3.4.31 isprime(x, {flag = 0}): true (1) if x is a (proven) prime number, false (0) otherwise. This can be very slow when x is indeed prime and has more than 1000 digits, say. Use ispseudoprime to quickly check for pseudo primality. See also factor.

If flag = 0, use a combination of Baillie-PSW pseudo primality test (see ispseudoprime), Selfridge "p − 1" test if x − 1 is smooth enough, and Adleman-Pomerance-Rumely-Cohen-Lenstra (APRCL) for general x.

[paulunderwood]

@James,

M82939 cofactor is certified prime

[Jean Penné]

Originally Posted by paulunderwood

@James,

M82939 cofactor is certified prime

Many congrats, Paul!

Jean

P.S. : How did you do the PRP test before the certification using Primo ?

[paulunderwood]

Thanks, Jean.

I merely got the candidate from www.mersenne.ca. I might have run a 3-PRP to be sure-ish. Anyway, Primo does a quick Fermat+Lucas à la BPSW before embarking on a lengthy ECPP path.

[Jean Penné]

Originally Posted by paulunderwood

Thanks, Jean.

I merely got the candidate from www.mersenne.ca. I might have run a 3-PRP to be sure-ish. Anyway, Primo does a quick Fermat+Lucas à la BPSW before embarking on a lengthy ECPP path.

Thank you for this detail!

Jean

[James Heinrich]

Originally Posted by paulunderwood

M82939 cofactor is certified prime

I have updated my PRP list, thanks.

[R. Gerbicz]

Originally Posted by Jean Penné

P.S. : How did you do the PRP test before the certification using Primo ?

Originally Posted by paulunderwood

Thanks, Jean.

I merely got the candidate from www.mersenne.ca. I might have run a 3-PRP to be sure-ish. Anyway, Primo does a quick Fermat+Lucas à la BPSW before embarking on a lengthy ECPP path.

We had already a Prp-cf test on this:
https://www.mersenne.org/report_expo...exp_hi=&full=1

Notice that for N=(k*2^n+c)/d we're using a Fermat test using
base^d as base, then
(base^d)^N=base^d mod N should hold for a prp number. So

base^(k*2^n+c)==base^d mod N, to help a lot we're using reduction mod (d*N)=mod (k*2^n+c).
Then do only one big division at the end of the test, in real life d is "small", at most ~1000 bits. And you can build in a strong check in the routine like for the normal prp test for k*2^n+c numbers. There is only a very small slow down at error check, because here our base is "large".

ps. so actually p95 has done a Fermat test using 3^d as base, and not 3. The reason is that we have a check only for 3^d [or base^d].