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- W4317464955 abstract "The classical Diophantine problem of determining which integers can be written as a sum of two rational cubes has a long history; from the earlier works of Sylvester, Selmer, Satg{'e}, {Lieman} etc. and up to the recent work of Alp{o}ge-Bhargava-Shnidman. 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." @default.
- W4317464955 created "2023-01-20" @default.
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- W4317464955 date "2023-01-17" @default.
- W4317464955 modified "2023-09-23" @default.
- W4317464955 title "Binary Cubic Forms and Rational Cube Sum Problem" @default.
- W4317464955 doi "https://doi.org/10.48550/arxiv.2301.06970" @default.
- W4317464955 hasPublicationYear "2023" @default.
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