A Conjecture Congenetic with Fermat’s Last Theorem

We propose the conjecture that for any positive integers r and n with n > 2, there do not exist 2r + 1 consecutive positive integers in natural order such that the sum of n-th powers of the first r + 1 integers equals the sum of n-th powers of the subsequent r integers, i.e., there are no positive integers r, m and n, where r < m and n > 2, satisfying (m – r)n + (m – r + 1)n + … + mn = (m + 1)n + (m + 2)n + … + (m + r)n. We prove that the conjecture is true for the cases n = 3 and n = 4. We also verified by using Mathematica that the conjecture is true for the cases 3 < n < 10 and m < 5000.