AI 中文总结
本文通过计算机辅助方法，结合Fano预解式、模性下降和有限Hecke比较，完整证明了广义费马方程x^3+y^5=z^7无非零互素整数解，并提供了可验证的精确计算证书。
AI 中文摘要
我们证明了广义费马方程 $x^3+y^5=z^7$ 在非零互素整数中无解，并明确说明了该证明如何扩展了下方引用的先前工作。Dahmen-Siksek 已建立了带符号的局部下降，并消除了关联的七次代数可约的三种情形。Putz 随后证明了剩余的不可约情形只能涉及七个七次域：六个纯域和一个例外域。缺失的步骤是排除这七个域。对于六个纯域，我们构造了一个显式的 Fano 预解式，下降到一个在 $\mathbb{Q}(\sqrt{-7})$ 上的亏格三的光滑平面四次曲线，并证明其有理参数轨迹仅由分支值上方的五个点组成。对于例外域，我们结合了 Pacetti-Villagra Torcomian 的模性和导子结果，以及在 $\mathbb{Q}(\sqrt{5})$ 上的水平降低和在 29 处的有限 Hecke 比较。我们还重构了 Dahmen-Siksek 的三个可约扇区论证作为可独立重放的计算，包括他们对二次域的无数据库识别，并在分歧素数处添加了一个内在的正式群处理。本文将所有主要步骤标记为文献输入、重构输入或新论证。所有项目特定的有限计算均以精确证书、程序、输入和认证日志的形式提供。
英文摘要
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 $\mathbb{Q}(\sqrt{-7})$, 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 $\mathbb{Q}(\sqrt{5})$ 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.
Comments43 pages. Computational companion (programs, exact inputs, certificates, verification records) and the record of an independent-machine replay: https://github.com/bbpcho/primitive-357, release replay-companion-2026-09-23.1. Substantial AI-assisted research and writing; see the statement in Section 8
