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

• All integer solutions of sums of M consecutive cubes equaling squares are found.
• Method of Pell equations yields recurrent solutions using Chebyshev polynomials.
• However, integer solutions are not ordered by increasing values of M.
• Solutions always exist for M odd, for M equal twice an integer square,
• and for first term in the sums equal odd integer squares.

All integer solutions (M,a,c)(M,a,c) to the problem of the sums of M   consecutive cubed integers (a+i)3(a+i)3 (a>1a>1, 0≤i≤M−10≤i≤M−1) equaling squared integers c2c2 are found by decomposing the product of the difference and sum of the triangular numbers of (a+M−1)(a+M−1) and (a−1)(a−1) in the product of their greatest common divisor g   and remaining square factors δ2δ2 and σ2σ2, yielding c=gδσc=gδσ. Further, the condition that g   must be integer for several particular and general cases yields generalized Pell equations whose solutions allow to find all integer solutions (M,a,c)(M,a,c) showing that these solutions appear recurrently. In particular, it is found that there always exists 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.