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

Abstract

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 $M\equiv 0,9,24$; or $M\equiv 1,2$ or $16\left(mod24\right)$; or $M\equiv 11\left(mod12\right)$. 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.

References in corpus (1)

• On the remarkable properties of the pentagonal numbers

Cited by in corpus (2)

• Finding all squared integers expressible as the sum of consecutive squared integers using generalized Pell equation solutions with Chebyshev polynomials
• 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