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math.NT — Number Theory

The generalized Fermat equation $x^2+y^5=z^7$

Contributed by Manvir Jaswal

We show that the equation $x^2+y^5=z^7$ has no solution in nonzero coprime integers, assuming a classification theorem from Putz's thesis, which has not appeared in a refereed journal. Equivalently, none of the equations $x^2+y^5=z^7$ , $x^2+y^7=z^5$ , $x^5+y^7=z^2$ has a solution in nonzero coprime integers. Putz attaches to a solution an octic algebra, the fibre of a Belyi map of degree $8$ , and proves that it is always isomorphic to one fixed octic field $L_8$ . We prove unconditionally that no solution has fibre isomorphic to $L_8$ , by a fifth-power descent over a field $L_{24}$ of degree $24$ that Putz introduced, completed by fifth-power residue symbols at five auxiliary primes. We also prove, without assuming the generalized Riemann hypothesis, that $L_{24}$ has class number $1$ , using a criterion of Belabas, Diaz y Diaz and Friedman.

Primarily AI-generated textHuman understanding: no parts

Oct 08, 2026PDF ↗View work →

hexagon:2610.00137v1

math.NT — Number Theory

Proof of the Non-existence of Perfect Cuboids via Mordell-Weil Rank Exhaustion and Minimal Polynomial Irreducibility of the Perfect Cuboid Surface

Contributed by Jonathan 𝑓(n) Reed

This manuscript establishes the non-existence of the Perfect Cuboid---a rectangular parallelepiped with integer edges, face diagonals, and space diagonal. By performing a rational sectioning of the governing quadratic forms, we demonstrate that the problem reduces to finding a non-trivial rational point on a family of hyperelliptic curves of Genus 3. We prove that the Jacobian of these curves possesses a Mordell-Weil rank of zero and that the perfection locus is an irrational algebraic singularity of degree $d = 4$ , precluding any solution in the integer domain $\mathbb{Z}^3$ . The non-existence of rational solutions is further verified via formal methods in Lean 4, demonstrating that the intersection of the Mordell-Weil torsion set and the degree-4 perfection locus is empty.

A mix of human-written and AI-generated textHuman understanding: all parts2-DescentEuler BrickHyperelliptic CurvesInteractive Theorem ProvingJacobian VarietyLean 4Mordell-Weil RankPerfect CuboidQuadratic Residueformal verification

Oct 07, 2026PDF ↗View work →

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The generalized Fermat equation $x^2+y^5=z^7$

Proof of the Non-existence of Perfect Cuboids via Mordell-Weil Rank Exhaustion and Minimal Polynomial Irreducibility of the Perfect Cuboid Surface

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Matching works

The generalized Fermat equation x2+y5=z7x^2+y^5=z^7x2+y5=z7

Proof of the Non-existence of Perfect Cuboids via Mordell-Weil Rank Exhaustion and Minimal Polynomial Irreducibility of the Perfect Cuboid Surface

The generalized Fermat equation $x^2+y^5=z^7$