Competition problem — Number theory

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Number theory Difficulty 5.0 AIME Find the answer

Compute the sum of all integers $1 \leq a \leq 10$ with the following property: there exist integers $p$ and $q$ such that $p, q, p^{2}+a$ and $q^{2}+a$ are all distinct prime numbers.

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Odd $a$ fail for parity reasons and $a \equiv 2(\bmod 3)$ fail for $\bmod 3$ reasons. This leaves $a \in\{4,6,10\}$ . It is easy to construct $p$ and $q$ for each of these, take $(p, q)=(3,5),(5,11),(3,7)$ , respectively.

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