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The generalized Fermat equation x3+y3=znx^3+y^3=z^n

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Contributed by Manvir Jaswal

Version 2 / Oct 07, 2026 / CC BY 4.0

Abstract

We prove that the equations x3+y3=znx^3+y^3=z^n and x3+y3=3znx^3+y^3=3z^n have no solution in coprime nonzero integers for any n≥3n\ge3. For the first equation the open cases were prime exponents p≡1(mod3)p\equiv1\pmod 3 above 10910^9 outside a set of congruence classes treated by Chen and Siksek; in these cases the solution 13+23=321^3+2^3=3^2 blocks the modular method. We replace Kraus's Frey curve by a hypergeometric motive of rank three over K=Q(−3)K=\mathbb{Q}(\sqrt{-3}), whose parameter at a solution is 33-adically close enough to a point of maximally unipotent monodromy for inertia at −3\sqrt{-3} to act unipotently; the parameter of 13+23=321^3+2^3=3^2 is not. By a theorem of Calegari, Emerton and Gee, the mod pp representation of the fiber, twisted by a cubic character, is the reduction of a member of a compatible system of minimal ramification, and we show that the member at a prime above 33 is irreducible and ordinary at −3\sqrt{-3}. Ramification and unconditional discriminant bounds show that its reductions have trivial semisimplification, and as the Galois group of the maximal pro-33 extension of KK unramified outside −3\sqrt{-3} is generated by one inertia group, the member stabilizes its ordinary line, a contradiction. For x3+y3=3znx^3+y^3=3z^n the same argument also works at the primes p≡2(mod3)p\equiv2\pmod3, which are inert in KK, and Frey curves and descents treat the remaining exponents.

Provenance statement

The author designed and operated the multi-agent research system used in this work, built on Anthropic Claude Opus 5.5. The system carried out literature analysis, mathematical development and computation, with checks assigned to agents separate from those developing the arguments. Claude Opus 5.5 was used to draft and revise the manuscript. Supplement (8 pp.): https://doi.org/10.5281/zenodo.23096423. Programs and data: https://doi.org/10.5281/zenodo.23096421.

Tools used

Anthropic
Claude OpusVersion 5.5
SageMath
SageMathVersion 10.9
PARI Group
PARI/GPVersion 2.17.3
GAP Group
GAPVersion 4.15.1
Python Software Foundation
Python (python-flint/Arb, SymPy, mpmath)Version 3.12

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Version history

  1. v1Submitted by Manvir JaswalInitial depositSupersededOct 07, 2026
  2. v2Submitted by Manvir JaswalSubmitted as a new versionCurrentOct 09, 2026