Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers

TL;DR

This paper investigates the conditions under which sums of consecutive squared integers equal a perfect square, establishing specific modular restrictions and characterizations for the number of terms involved.

Contribution

It provides a comprehensive set of congruence conditions and characterizations for the number of terms in sums of consecutive squares that result in perfect squares, including special cases where the number of terms is a perfect square.

Findings

No solutions for certain mod 12 classes of M.

Solutions exist for specific mod 72 and mod 24 classes of M.

When M is a perfect square, it relates to pentagonal numbers.

Abstract

Considering the problem of finding all the integer solutions of the sum of $M$ consecutive integer squares starting at $a^{2}$ being equal to a squared integer $s^{2}$, it is shown that this problem has no solutions if $M\equiv3,5,6,7,8$ or $10 (mod\,12)$ and has integer solutions if $M\equiv0,9,24$ or $33 (mod\,72)$; or $M\equiv1,2$ or $16 (mod\,24)$; or $M\equiv11 (mod\,12)$. All the allowed values of $M$ are characterized using necessary conditions. If $M$ is a square itself, then $M\equiv1 (mod\,24)$ and $(M-1)/24$ are all pentagonal numbers, except the first two.