Track / Stage 4 / 315 of 340 #575 of 1964

Problem 575

AMC 12 late, AIME early

Number theory Difficulty 5.0 Prove it Thai Mathematical Olympiad · Thailand

Let $n$ be a positive integer for which $5n + 1$ is a perfect square. Show that $n + 1$ is a sum of 5 perfect squares.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Let $5n + 1 = m^2 \equiv 1 \pmod{5}$ . Thus $m = 5k \pm 1$ for some integer $k$ . We have
$n + 1 = \frac{(5k \pm 1)^2 + 4}{5} = 5k^2 \pm 2k + 1 = 4k^2 + (k \pm 1)^2$
which can be written as a sum of 5 perfect squares as desired.

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