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

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

On Sun, Nov 17, 2024 at 04:15:21PM +0100, hermann@stamm-wilbrandt.de wrote:
> 
> > On 2024-11-17 03:08, Kurt Foster wrote:
> > > In any case, the prime M splits into two prime ideals in Q(sqrt(-23)).
> > >  If M = x^2 + 23*y^2 the ideals are principal; otherwise not.
> > > 
> In addition to the questions from my previous email,
> are you sure that this applies to Mersenne primes as well?
> 
> I tested the Mersenne exponents with -23 being quadratic
> residue up to 607 manually in gp, see below.
> 
> And for M_18..M_38 with -23 being quadratic residue with automation, below.
> 
> For all 18 primes sqrt(Mod(-23,p)) got computed and verified, not a single
> exception ...

Even if the squareroot exists, the equation M = a^2+23*b^2 might not
have a solution.
However in that case the equation M = 2*a^2+a*b+3*b^2 must have a solution!

Cheers,
Bill.