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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 thedifference 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 $g$ andremaining square factors $\delta^{2}$ and $\sigma^{2}$, yielding$c=g\delta\sigma$. Further, the condition that $g$ must be integer for severalparticular and general cases yield generalized Pell equations whose solutionsallow to find all integer solutions $\left(M,a,c\right)$ showing that thesesolutions appear recurrently. In particular, it is found that there alwaysexist at least one solution for the cases of all odd values of $M$, of all oddinteger square values of $a$, and of all even values of $M$ equal to twice aninteger square.

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