General solutions of sums of consecutive cubed integers equal to squared integers

TL;DR

This paper derives a general method to find all integer solutions where sums of consecutive cubed integers equal perfect squares, revealing recurring solutions and specific conditions for existence based on Pell equations.

Contribution

It introduces a novel approach using triangular number decompositions and Pell equations to systematically find solutions for sums of consecutive cubes equaling squares.

Findings

Solutions exist for all odd M, odd square a, and even M equal to twice a square.

Solutions recur and can be characterized by Pell equations.

A general formula for solutions is provided.

Abstract

All integer solutions $\left(M,a,c\right)$ to the problem of the sums of $M$ consecutive cubed integers $\left(a+i\right)^{3}$ ($a>1$, $0\leq i\leq M-1$) equaling squared integers $c^{2}$ are found by decomposing the product of the difference and sum of the triangular numbers of $\left(a+M-1\right)$ and $\left(a-1\right)$ in the product of their greatest common divisor $g$ and remaining square factors $\delta^{2}$ and $\sigma^{2}$, yielding $c=g\delta\sigma$. Further, the condition that $g$ must be integer for several particular and general cases yield generalized Pell equations whose solutions allow to find all integer solutions $\left(M,a,c\right)$ showing that these solutions appear recurrently. In particular, it is found that there always exist at least one solution for the cases of all odd values of $M$, of all odd integer square values of $a$, and of all even values of $M$ equal to twice an integer square.