Integer sequences of the form α^n±β^n

Abstract

Suppose that we want to find all integer sequences of the form α^n + β^ n , where α and β are complex numbers and n is a nonnegative integer. Since α^0 + β ^0 is always an integer, our task is then equivalent to determining all complex pairs (α, β) such that α^n + β^n ∈ Z, n > 0.