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 12:32:44PM +0100, tony.reix@laposte.net wrote:
> ‌Hi Hermann,
> 
> Reading your email on my phone, that looks interesting. However, I'll need to read it on my PC then.
> 
> 
> 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).
> 
>  
> Experimenting quickly with Wolfram Alpha, it seems that the Mersenne primes are:
>       M_q = 4x^2+3y^2

This is true for every prime numbers congruent to 7 mod 12

> and often:
>       M_q = 4x^2+27y^2 .

This ones requires additionaly that 2 is a cube mod p by a well-known
result of Gauss.
However this condition is always true for Mersenne primes for q > 3

Indeed let q > 3 such that M_q is prime
M_q = 2^q-1
since q is odd:   M_q = 1 (mod 3)
since q is prime: M_q = 1 (mod q)

so M_q-1 = 3*q*u for some integer u.

2^((M_q-1)/3) = (2^q)^u = 1 [mod M_q]

so 2 is a cube mod M_q and so M_q can be written as 4x^2+27y^2.

Cheers,
Bill