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

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.