Track / Stage 4 / 218 of 340 #478 of 2000

Problem 478

AMC 12 late, AIME early

Number theory Difficulty 4.9 Find the answer

2 [

Find all pairs of prime numbers, the difference of the squares of which is a prime number.

Your answer

Official solution

Let $p$ and $q$ be prime numbers and $p^{2}-q^{2}=(p-q)(p+q)$ be a prime number. Then $p-q=1$ . Therefore, one of our numbers is even, that is, $q=2, p=3$ .

## Answer

$(3,2)$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.