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arXiv · 2301.06970

Binary Cubic Forms and Rational Cube Sum Problem

Somnath Jha·Dipramit Majumdar·B. Sury
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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.

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Somnath Jha, Dipramit Majumdar, B. Sury. 2023-01-17. Binary Cubic Forms and Rational Cube Sum Problem. https://arxiv.org/abs/2301.06970

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