Re: PARI/GP timings for operations on biggest known 41,024,320 decimal digit prime

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Sat, Nov 16, 2024 at 04:48:44PM +0100, hermann@stamm-wilbrandt.de wrote:
> On 2024-11-16 12:32, tony.reix@laposte.net wrote:
> > BTW, do you know that Mersenne numbers can also be written as:
> >    M_q=(8x)^2-(3qy)^2
> > ?
> > Once if M_q is prime (and x & y are obvious). Several times if M_q is
> > composite (and x & y are not obvious at all).
> > 
> Interesting, the questions you had regarding this were answered by Bill.
> 
> > Do you have a link to a paper clarifying the property of M_q as
> > x^2+3y^2 ?
> > 
> First, for me M_n is the n-th Mersenne prime, and not 2^n-1.
> 
> All known 52 Mersenne primes but M_1 have -3 as quadratic residue:

This is true for all Mersenns primes except M_1.

2^p-1 = (-1)^p-1 mod 3 
since p is odd 
(-1)^p = -1 so 
2^n-1 = -2 (mod 3) = 1 (mod 3) 
so -3 is quadratic residue mod 2^p-1.

Cheers,
Bill.