Re: Pell's equations and beyond

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

On Tue, Dec 19, 2023 at 04:04:04PM +0100, Denis Simon wrote:
> Hi,
> 
> If norm(u) = 1, is there a quick way to test whether this u is a square or not ?

The thing is, you do not want to actually compute u explicitely,
However you can easily compute u mod p for a lot of p.

If you find a prime number so that u is not a square mod p, this proves
that u is not a square (and you also need to prove that -u is not a square
to prove there is no units of norm -1).

Now if you do not find such p small enough, you can prove that GRH is false,
but you still does not know whether u is a square or not.

Cheers,
Bill.