Track / Stage 4 / 135 of 340 #395 of 1964

Problem 395

AMC 12 late, AIME early

Number theory Difficulty 4.8 Find the answer

Find all integers $n$ such that $4 n^{4}+1$ is a prime number.

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

By the Sophie-Germain identity:

$1+4 n^{4}=\left(1-2 n+2 n^{2}\right)\left(1+2 n+2 n^{2}\right)$ Therefore, it is necessary that $1-2 n+2 n^{2}=1$ , so $n=1$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.