Equations of two sets of consecutive square sums

Abstract

In this paper we investigate equations featuring sums of consecutive square integers, such as $3^2 + 4^2 = 5^2$, and $108^2 + 109^2 + 110^2 = 133^2 + 134^2$. In general, for a sum of $m+1$ consecutive square integers, $x^2 + (x+1)^2 + \cdots + (x+m)^2$, there is a distinct set of $m$ consecutive squares, $(x+n)^2 + (x+(n+1))^2 + \cdots + (x+(n+(m-1)))^2$, to which these are equal. We present a bootstrap method for constructing these equations, which yields solutions comprising an infinite two-dimensional array. We apply a similar method to constructing consecutive square sum equations involving $m+2$ terms on the left, and $m$ terms on the right, formed from two distinct sets of consecutive squares separated one term to the left of the equals sign, such as $2^2 + 3^2 + 6^2 = 7^2$.