Characterizing the sum of two cubes

Abstract

An intrinsic characterization of positive integers which can be represented as the sum or difference of two cubes is given. Every integer has a smallest multiple with is a sum of two cubes and such that the multiple, in the form of an iterated function of the integer, is eventually periodic with period one or two. The representation of any integer as the sum of two cubes to a fixed modulus is always possible if and only if the modulus is not divisible by 7 or 9.