M971 factored

[philmoore]

Congratulations to the factorer of M971, which had been the smallest Mersenne number without a known (prime) factor. Does anyone have any information about this? The original number had 293 digits, and a 53-digit factor was sufficient to crack it. The smallest known unfactored Mersenne distinction now goes to M1061 at 320 digits. See the update at:
http://www.mersenne.org/ecmm.htm

[Uncwilly]

It says that factorization is complete on it too.

[akruppa]

Very nice indeed! Congrats to the unknown factorer!

Any info on what the sigma of the lucky curve was?

Alex

[xilman]

Originally Posted by akruppa

Very nice indeed! Congrats to the unknown factorer!

Any info on what the sigma of the lucky curve was?

Alex

I and a bunch of others (including Alex, I now see) received this mail from George Woltman overnight. It answers the questions asked so far in this thread.

Cc: David Symcox [Removed email address - Mod]
Subject: Fwd: M971 has a factor: 23917104973173909566916321016011885041962486321502513
Date: Mon, 13 Sep 2004 16:18:20 -0400
Hello all,

Congratulations to David for finding a 53-digit factor of M971 !!
Proving the cofactor prime is left as an exercise to the reader....

...

> [Tue Sep 07 23:37:21 2004]
> ECM found a factor in curve #1356, stage #2
> Sigma=7917157863480469, B1=44000000, B2=4290000000.
> M971 has a factor: 23917104973173909566916321016011885041962486321502513
> Cofactor is a probable prime!

Paul

[akruppa]

Oh, the email went to my old address which I don't read as often anymore (swamped by spam).

The group order, thanks to the Magma calculator is 2^4 * 3 * 29 * 31^2 * 41 * 53 * 107 * 21589 * 62773 * 140419 * 229781 * 238729 * 2286601 * 3221525051, computed with

FindGroupOrder2 := function (p, s)
   K := GF(p);
   v := K ! (4*s);
   u := K ! (s^2-5);
   x := u^3;
   b := 4*x*v;
   a := (v-u)^3*(3*u+v);
   A := a/b-2;
   x := x/v^3;
   b := x^3 + A*x^2 + x;
   E := EllipticCurve([0,b*A,0,b^2,0]);
   return FactoredOrder(E);
end function;
p:=23917104973173909566916321016011885041962486321502513;
s:=7917157863480469;
FindGroupOrder2(p,s);

Unfortunately, the factor is just a little too small for Brent's ECM Top Ten.

Alex

[Uncwilly]

Originally Posted by no one in particular

M971 has a factor: 23917104973173909566916321016011885041962486321502513
Cofactor is a probable prime!

Using Dario's applet, the cofactor is indeed prime. That means that M971 is the product of 2 primes.