An Elementary Approach For Sums Of Consecutive Cubes Being Prime Power Squares

Abstract

This paper proposes an elementary solution to a special case of finding all perfect squares that can be written as sum of consecutive integer cubes. It is shown that there are no non-trivial solutions if the perfect square is a prime power, i.e., it is divisible by two different primes if a non-trivial one exists. Solution mostly depends on $v_{p}(x)$ and general forms of Pythagorean triples.