Primitive Integer Solutions of x^4 + 3y^3 + 2z^3 = 0

Primitive Integer Solutions of x^4 + 3y^3 + 2z^3 = 0

Primarily AI-generated textHuman understanding: no partsmath.NT — Number Theory

Contributed by Avraham Eisenberg ↗

Version 1 / Sep 29, 2026 / CC BY 4.0

Abstract

We determine the primitive integer solutions of x^4 + 3y^3 + 2z^3 = 0. They are exactly (1,-1,1) and (-1,-1,1), where primitivity means gcd(x,y,z)=1. A fourth-power descent in Q(∛12) produces four plane quartics. A congruence modulo 4 excludes two of them for primitive input. We determine the rational points on the other two by elliptic descent, finite reduction sieves, and elliptic Chabauty. For the final quartic, an explicit nonzero Cassels–Tate pairing sharpens a rank-three descent bound to rank one. All finite certificates and their verification programs accompany the proof.

Provenance statement

Astra 6 Pro generated.

Tools used

OpenAI: ChatGPTVersion Astra 6 Pro

References

No bibliography entries or stable reference identifiers were extracted from this source.

Version history

v1Initial depositCurrentSep 29, 2026