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

摘要

All integer solutions (M, a, c) to the problem of the sums of NI consecutive cubed integers (a + i)(3) (a > 1, 0 <= i <= M 1) equaling squared integers c(2) are found by decomposing the product of the difference and sum of the triangular numbers of (a + M - 1) and (a - 1) 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 yields generalized Pell equations whose solutions allow to find all integer solutions (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. (C) 2015 Elsevier Inc. All rights reserved.