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

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.