Decomposition of an integer as the sum of two cubes to a fixed modulus

TL;DR

This paper investigates conditions under which integers can be expressed as the sum of two cubes modulo a fixed number, providing an inductive method to find such representations and their components.

Contribution

It introduces a criterion for representability as the sum of two cubes modulo N and presents an inductive approach to determine these representations for non-prime N.

Findings

Representation is possible iff the modulus is not divisible by 7 or 9.

An inductive method to find remainders that are sums of two cubes.

Explicitly finds the components of the sum of two cubes representations.

Abstract

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 seven or nine. For a positive non-prime integer N there is given an inductive way to find its remainders that can be represented as the sum of two cubes to a fixed modulus N. Moreover, it is possible to find the components of this representation.