Primes x^2+m y^2

[Max Alekseyev]

On Thu, Apr 24, 2008 at 1:16 AM, Artur <grafix at csl.pl> wrote:

> Who can explain me why primes of the form x^2+5 y^2 are that same as x^2+25 y^2

That's not true: 5 = 0^2 + 5 * 1^2 but 5 is not of the form x^2+25 y^2.
However, this is the only exception (and that's quite surprising indeed).

Check out the theorem on primes of the form x^2 + n*y^2 at the link
I've gave earlier today.
Basically, to be represented as x^2+25 y^2 the prime p>5 should
satisfy (-5/p)=1 and (5/p)=1, implying that p = 1 or 9 modulo 20. And
the same is required (in the equivalent form: (-1/p)=1 and (5/p)=1) to
be represented as x^2 + 25y^2.

> but x^2+7 y^2 are different as x^2+49 y^2  ?

But this is not surprising at all. One possible indication is
inequality of the class numbers: h(-4*7)=1 and h(-4*49)=4.

Regards,
Max