On sums of powers of integers

Abstract

For non-negative integers k, n, let Pk(n) denote the sum {fx27-1}. We show by two different means that if k ≥ 3 and odd, then n2(n+1)2 iss a factor of the polynomial Pk(n); and if k ≥ 2 and even, then n (n+1) (2n+1) is a factor of the polynomial Pk(n). We also derive a relatively unknown result first obtained by Johann Faulhaber in the 17th century.