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

The problem of finding all the integer solutions in $a$, $M$ and $s$ of sums of $M$ consecutive integer squares starting at $a^2\ge 1$ equal to squared integers $s^2$, has no solutions if $M\equiv 3,5,6,7,8$ or $10\left(mod12\right)$ 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.