Binary Cubic Forms and Rational Cube Sum Problem

TL;DR

This paper uses integral binary cubic forms to demonstrate that infinitely many primes in specific residue classes are sums of two rational cubes, advancing understanding of the rational cube sum problem.

Contribution

It establishes unconditionally that infinitely many primes in certain residue classes are sums of two rational cubes, and explores related prime distribution results.

Findings

Infinitely many primes in residue classes 1 mod 9d and -1 mod 9d are sums of two rational cubes.

Every non-zero residue class mod q contains infinitely many primes that are sums of two rational cubes.

Existence of infinitely many primes p in classes 8 mod 9 and 1 mod 9 such that Np is a sum of two rational cubes.

Abstract

In this note, we use integral binary cubic forms to study the rational cube sum problem. We prove (unconditionally) that for any positive integer $d$, infinitely many primes in each of the residue classes $ 1 \pmod {9d}$ as well as $ -1 \pmod {9d}$, are sums of two rational cubes. Among other results, we prove that every non-zero residue class $a \pmod {q}$, for any prime $q$, contains infinitely many primes which are sums of two rational cubes. Further, for an arbitrary integer $N$, we show there are infinitely many primes $p$ in each of the residue classes $ 8 \pmod 9$ and $1 \pmod 9$, such that $Np$ is a sum of two rational cubes.