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arXiv · 1409.7969

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

Vladimir Pletser
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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.

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Vladimir Pletser. 2014-09-28. Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers. https://arxiv.org/abs/1409.7969

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