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Problem 1068

AMC 12 late, AIME early

Number theory Difficulty 5.0 Find the answer HMMT February

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. Fractions can be typed as 3/2, and spacing doesn't matter.

Official 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.

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.