Problem

Source: TST 3,day 1,P 3

Tags: number theory, prime numbers, modular arithmetic, Quadratic Residues



Let $p \neq 13$ be a prime number of the form $8k+5$ such that $39$ is a quadratic non-residue modulo $p$. Prove that the equation $$x_1^4+x_2^4+x_3^4+x_4^4 \equiv 0 \pmod p$$has a solution in integers such that $p\nmid x_1x_2x_3x_4$.