Shifted Power of a Polynomial with Integral Roots

Abstract

ABSTRACT In this note, for any integers m, n ≥ 2, we find a condition on a positive integer c under which there exists a monic polynomial f ∈ ℤ [ x ] of degree n for which f(x) m – c has mn integral roots counting with multiplicities. This is the case if and only if m = 2 and c is a constant that comes from a solution of the Prouhet-Tarry-Escott problem of size n. For example, the smallest positive integer c for which there exists a monic degree 7 polynomial f ∈ ℤ [ x ] such that f(x)2–c has 14 integral roots is c = 6620176679276160000.