Track / Stage 5 / 399 of 400 #999 of 1964

Problem 999

AIME late

Number theory Difficulty 6.0 Prove it

13. Let $p$ be an odd prime. Prove: The indeterminate equation $p=x^{2}+2 y^{2}$ has a solution if and only if $\left(\frac{-2}{p}\right)=1$ , i.e., $p \equiv 1$ or $3(\bmod 8)$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

13. The proof is the same as Theorem 4.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.