Representations of some primes as $x^2-2y^2$?

Question

I am looking for (elementary) proofs (idea of the proofs is also OK) or references to proofs of the followings:

$$ p\equiv\pm1(\mod8)\longrightarrow p=x^2-2y^2 $$

Any help appreciated.

Answer 1

This is the general idea.

Pigeonhole Principle

Note, that since $k$ is a quadratic residue, this implies there exists such $a$ that $a^2 \equiv k \pmod p$.

By Thue's Lemma, we get that there exists such $-\sqrt{p}< x,y < \sqrt{p}$ that $x \equiv ay \pmod {p}$.

EXAMPLE

Assume that $p \equiv \pm 1 \pmod 8.

Note, that since $2$ is a quadratic residue, this implies there exists such $a$ that $a^2 \equiv 2 \pmod p$.

By Thue's Lemma, we get that there exists such $-\sqrt{p}< x,y < \sqrt{p}$ that $x \equiv ay \pmod {p}$.

This implies that $x^2-a^2y^2 \equiv x^2-2y^2 \equiv 0 \pmod {p}$.

Note that $-2p<x^2-2y^2<p$.

This implies that that $p=2y^2-x^2$ which is equivalent to saying $p=(2y+x)^2-2(x+y)^2$.