Infinitely many primes in each of the residue classes $1$ and $8$ modulo $9$ are sums of two rational cubes

Somnath Jha, Dipramit Majumdar, B. Sury

Published 2023-01-17Version 1

Given an integer $n$, it is a classical Diophantine problem to determine whether $n$ can be written as a sum of two rational cubes. The study of this problem 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{\"o}ge-Bhargava-Shnidman. In this short note, we prove that infinitely many primes congruent to $1$ modulo $9$ are sums of two rational cubes and that infinitely many primes congruent to $8$ modulo $9$ are sums of two rational cubes. Among other results, we prove also that every residue class $a \pmod {q}$ (for any prime $q$ and any integer $a$ co-prime to $q$) contains infinitely many primes which are sums of two rational cubes. Further, we provide many infinite families of composite numbers which are rational cube sums.