On the Sum of Consecutive Squares

Abstract

We seek solutions of the Diophantine equation

$$ {\left( {n + 1} \right)^2} + {\left( {n + 2} \right)^2} + \cdots + {\left( {n + k} \right)^2} = {m^2} $$

((1))

for integers n, k and m, with n ≥ − 1 and with k and m positive and, in §3, we will also briefly consider the related equation where m ² is replaced by m ³. When k = 1, (1) is trivial. For k = 2 we replace n by n − 1 and express (1) in the form

$$ {n^2} + {\left( {n + 1} \right)^2} = {m^2}. $$

((2))