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

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 $M\equiv 3,5,6,7,8$ or $10(mod12)$ and has integer solutions if $M\equiv 0,9,24$ or $33(mod72)$; or $M\equiv 1,2$ or $16(mod24)$; or $M\equiv 11(mod12)$. All the allowed values of $M$ are characterized using necessary conditions. If $M$ is a square itself, then $M\equiv 1(mod24)$ and $(M-1)\mathrm{/}24$ are all pentagonal numbers, except the first two.