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

TL;DR

This paper investigates specific congruence conditions on the parameters governing sums of consecutive squared integers that equal perfect squares, revealing detailed modular relationships for these sums and their initial terms.

Contribution

It establishes new congruence conditions on parameters like M, η, δ, and the difference of initial terms for sums of consecutive squares equaling squares, extending previous understanding.

Findings

η is odd, i.e., η ≡ 1 mod 2

δ can be 0, 1, or 5 mod 6, with specific conditions on M

Initial terms a₁ and a₂ have different or same parity depending on δ

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.