The Primitive Generalized Fermat Equation x^3+y^5=z^7: A computer-assisted proof
Abstract
We prove that the generalized Fermat equation x3+y5=z7 has no solution in nonzero coprime integers, and we make explicit how the proof extends the prior work cited below. Dahmen-Siksek had established the signed local descent and eliminated the three cases in which the associated degree-seven algebra is reducible. Putz had then proved that the remaining irreducible case can involve only seven septic fields: six pure fields and one exceptional field. The missing step was to exclude those seven fields. For the six pure fields we construct an explicit Fano resolvent, descend to a smooth plane quartic of genus three over ℚ(√(−7)), and prove that its rational-parameter locus consists only of five points above the branch values. For the exceptional field we combine the modularity and conductor results of Pacetti-Villagra Torcomian with level lowering over ℚ(√5) and a finite Hecke comparison at $29$. We also reconstruct the three reducible-sector arguments of Dahmen-Siksek as independently replayable calculations, including their database-free identification of the quadratic field, and add an intrinsic formal-group treatment at the ramified prime. The paper labels every major step as a literature input, reconstructed input, or new argument. All project-specific finite calculations are supplied as exact certificates, programs, inputs, and authenticated logs.
Related
- Primitive norm extraction and a signed descent for the cubic Fermat equation
- Generalized Fermat equation over number fields
- A Computer-Assisted Proof that R(5,10;5,8)=21 for the Generalized Threshold Ramsey Number
- Existence and spectral instability of steady rolls in the Nikolaevskiy equation: a computer-assisted study
- Fermat's Last Theorem in star-invariant subspaces II: small boundary spectrum, projective plane curves, and the cubic Fermat equation
BibTeX
@misc{chocian2026primitive,
title = {{The Primitive Generalized Fermat Equation x^3+y^5=z^7: A computer-assisted proof}},
author = {Peter Chocian},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26996},
eprint = {2609.26996},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26996},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Primitive Generalized Fermat Equation x^3+y^5=z^7: A computer-assisted proof
AU - Peter Chocian
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26996
UR - https://arxiv.org/abs/2609.26996
AB - We prove that the generalized Fermat equation x^3+y^5=z^7 has no solution in nonzero coprime integers, and we make explicit how the proof extends the prior work cited below. Dahmen-Siksek had established the signed local descent and eliminated the three cases in which the associated degree-seven algebra is reducible. Putz had then proved that the remaining irreducible case can involve only seven septic fields: six pure fields and one exceptional field. The missing step was to exclude those seven fields. For the six pure fields we construct an explicit Fano resolvent, descend to a smooth plane quartic of genus three over ℚ(√(−7)), and prove that its rational-parameter locus consists only of five points above the branch values. For the exceptional field we combine the modularity and conductor results of Pacetti-Villagra Torcomian with level lowering over ℚ(√5) and a finite Hecke comparison at $29$. We also reconstruct the three reducible-sector arguments of Dahmen-Siksek as independently replayable calculations, including their database-free identification of the quadratic field, and add an intrinsic formal-group treatment at the ramified prime. The paper labels every major step as a literature input, reconstructed input, or new argument. All project-specific finite calculations are supplied as exact certificates, programs, inputs, and authenticated logs.
ER -