Competition problem — Number theory

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Number theory Difficulty 6.3 National olympiad Prove it

14. Let $p>5$ be an odd prime. Is the statement "The indeterminate equation $p=x^{2}+5 y^{2}$ has a solution if and only if $\left(\frac{-5}{p}\right)=1$ " correct? Can the proof method of Theorem 4 in § 2 be used here? Why?

Solution

14. Not applicable, the last part of the sufficiency proof may not hold here. For example, when $p=7$ , $\left(\frac{-5}{7}\right)=1$ , but $7=x^{2}+5 y^{2}$ has no solution.

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