Let $a$ be a positive integer. Find all prime numbers $ p $ with the following property: there exist exactly $ p $ ordered pairs of integers $ (x, y)$, with $ 0 \leq x, y \leq p - 1 $, such that $ p $ divides $ y^2 - x^3 - a^2x $.
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Tags: SAU, Divisibility
Let $a$ be a positive integer. Find all prime numbers $ p $ with the following property: there exist exactly $ p $ ordered pairs of integers $ (x, y)$, with $ 0 \leq x, y \leq p - 1 $, such that $ p $ divides $ y^2 - x^3 - a^2x $.