Competition problem — Number theory

Library / /613 of 1394

Number theory Difficulty 5.2 AIME, harder Prove it United States

Problem:

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.

Solution

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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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.