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Binary Cubic Forms and Rational Cube Sum Problem

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arxiv 2301.06970 v4 pith:B7ZWZBRW submitted 2023-01-17 math.NT

Somnath Jha , Dipramit Majumdar , B. Sury This is my paper · ORCID
classification math.NT
keywords pmodrationalcubesinfinitelymanyprimesresiduebinary
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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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Cited by 3 Pith papers

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  1. A Ramanujan Pell Elliptic K3 Surface and Primitive Five-Cube Near Misses

    math.GM 2026-07 unverdicted novelty 6.0 of 10

    A Ramanujan–Pell construction yields primitive five-cube near-misses and an elliptic K3 of fibre type 6IV with a height-4/3 section and a visible rank-16 Néron–Severi sublattice of discriminant −108.

  2. Relative $p$-class groups and $p$-Selmer groups

    math.NT 2024-12 conditional novelty 5.0 of 10

    For CM elliptic curves with j-invariant 0 or 1728, the p-Selmer rank of a twist is determined by the root number and a χ-component of a relative p-class group.

  3. Hilbert's 10th Problem via Mordell curves

    math.NT 2024-12 conditional novelty 4.0 of 10

    For five-sixths of primes p, Hilbert's 10th problem is shown unsolvable over Q(ζ3, ∛p), with a companion result for degree-12 extensions Q(ζ3, √D, ∛p).

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