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

TL;DR

This paper refines the conditions under which sums of consecutive squared integers equal perfect squares by applying additional congruence constraints, building on Beeckmans' necessary conditions.

Contribution

It introduces new congruence conditions that further restrict the values of M for which the sum of M consecutive squares equals a perfect square, extending previous results.

Findings

Certain M values are excluded based on new congruence conditions.

Refined criteria narrow the search for solutions to specific residue classes.

The results improve understanding of the structure of solutions in sums of squares problems.

Abstract

The problem of finding all the integer solutions in $a$, $M$ and $s$ of sums of $M$ consecutive integer squares starting at $a^{2}\geq1$ equal to squared integers $s^{2}$, has no solutions if $M\equiv3,5,6,7,8$ or $10\left(mod\,12\right)$ and has integer solutions if $M\equiv0,9,24% or %33\left(mod\,72\right)$; or $M\equiv1,2$ or $16\left(mod\,24\right)$; or $M\equiv11\left(mod\,12\right)$. In this paper, additional congruence conditions are demonstrated on the allowed values of $M$ that yield solutions to the problem by using Beeckmans' eight necessary conditions, refining further the possible values of $M$ for which the sums of $M$ consecutive integer squares equal integer squares.