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

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

Contributed by Avraham Eisenberg ↗

SubmitterAvraham Eisenberg

Version 1 / Sep 29, 2026 / CC BY 4.0

Abstract

We determine all primitive integer solutions of the generalized Fermat equation x^4 + 4y^3 + z^3 = 0. Here primitive means gcd(x,y,z)=1. We prove that the only primitive solutions are (x,y,z) = (1,0,-1), (-1,0,-1). The proof begins with a fourth-power descent in the pure cubic field Q(∛2), reducing the problem to the rational points on two explicit smooth plane quartics of genus 3. These quartics admit maps to elliptic curves over Q(∛2). Exact 2-descent determines the relevant Mordell-Weil ranks and finite-index subgroups. A complete reduction sieve at 127 leaves four residue disks, and an elliptic Chabauty calculation proves that each disk contains exactly one rational point. Reconstruction of the original variables then leaves precisely the two stated primitive solutions. The finite computations are exact and reproducible from the certificates and verifier source included in the appendices, and the argument applies at all heights.

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References

  1. M. Stoll, Implementing 2-descent for Jacobians of hyperelliptic curves , Acta Arith. 98 (2001), 245--277, especially Lemmas 4.1--4.4.
  2. N. Bruin, Chabauty methods using elliptic curves , J. Reine Angew. Math. 562 (2003), 27--49.
  3. B. Grechuk, A systematic approach to Diophantine equations: open problems , arXiv:2404.08518v9 [math.GM], 30 August 2026, Table 4, p. 4. https://arxiv.org/abs/2404.08518v9 . DOI: https://doi.org/10.48550/arXiv.2404.08518 10.48550/arXiv.2404.08518 .arXivDOI

Version history

  1. v1Initial depositCurrentSep 29, 2026