2023/01/17 by Somnath Jha, Jha, Somnath, Dipramit Majumdar +3 · 2 citations
Physics and Astronomy · Mathematics · #Advanced Mathematical Theories and Applications #Analytic Number Theory Research
paper · pdf · doi:10.48550/arxiv.2301.06970
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.