Congruent conditions on the number of terms, on the ratio number of terms to first terms and on the difference of first terms for sums of consecutive squared integers equal to squared integers

Abstract

Sums of $M$ consecutive squared integers $\left(a+i\right)^{2}$ equaling squared integers (for $a\geq1$, $0\leq i\leq M-1$) yield certain linear groupings of pairs $\left(a_{1},a_{2}\right)$ of $a$ values for successive same values of $M$ when these are linked by $a_{1}+a_{2}=\mu M+1$ with $\mu=\left(\eta/\delta\right)\in\mathbb{\mathbb{Q}}^{+}$. In this paper, congruent conditions on $M,\eta,\delta$, and on the difference $\left(a_{2}-a_{1}\right)$ are demonstrated for these linear groupings to hold. It is found that $\eta\equiv1\left(mod\,2\right)$ and $\delta\equiv0,1$ or $5\left(mod\,6\right)$, and if $\delta\equiv0\left(mod\,6\right)$, $M\equiv0\left(mod\,12\right)$, while if $\delta\equiv1$ or $5\left(mod\,6\right)$, $M\equiv2$ or $11\left(mod\,12\right)$ with $a_{1}$ and $a_{2}$ being of different or same parities.