Cube sum problem for integers having exactly two distinct prime factors

TL;DR

This paper investigates the classical problem of representing certain integers as sums of two rational cubes, focusing on cube-free integers with exactly two distinct prime factors, expanding understanding of this longstanding Diophantine challenge.

Contribution

It specifically addresses the cube sum problem for a new class of integers, namely cube-free integers with two distinct primes none of which is 3, providing new insights and results.

Findings

Characterization of cube sum representations for the specified integers

Extension of previous results to a broader class of integers

New criteria or conditions for the sum of two rational cubes

Abstract

Given an integer n>1, it is a classical Diophantine problem that whether n can be written as a sum of two rational cubes. The study of this problem, considering several special cases of n, has a copious history that can be traced back to the works of Sylvester, Satg\'e, Selmer etc. and up to the recent works of Alp\"oge-Bhargava-Shnidman. In this article, we consider the cube sum problem for cube-free integers n which has two distinct prime factors none of which is 3.