Solving \(a x^p + b y^p = c z^p\) with \(abc\) containing an arbitrary number of prime factors

DOI10.1007/s00009-020-01678-1zbMath1478.11040arXiv1811.05957OpenAlexW3124926382MaRDI QIDQ2223085

Eduardo Soto, Luis V. Dieulefait

Publication date: 28 January 2021

Published in: Mediterranean Journal of Mathematics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1811.05957

zbMATH Keywords

\(S\)-unit equationasymptotic Fermat conjectureFermat equation with coefficients

Mathematics Subject Classification ID

Elliptic curves over global fields (11G05) Congruences for modular and (p)-adic modular forms (11F33) Galois representations (11F80) Exponential Diophantine equations (11D61) Higher degree equations; Fermat's equation (11D41) Cyclotomic extensions (11R18)

Uses Software

• LMFDB

Cites Work

• Dimensions of the spaces of cusp forms and newforms on \(\Gamma_0(N)\) and \(\Gamma_1(N)\)
• On Néron's classification of elliptic curves of residue characteristics 2 and 3
• Simplifying the solution of Ljunggren's equation \(x^ 2+1=2y^ 4\)
• Sur les représentations galoisiennes modulo \(\ell\) attachées aux formes modulaires. (On Galois representations modulo \(\ell\) attached to modular forms)
• Finiteness theorems for abelian varieties over number fields.
• On the modular representations of degree two of \(\text{Gal}({\overline {\mathbb Q}}/{\mathbb Q})\)
• Twists of newforms and pseudo-eigenvalues of \(W\)-operators
• Rational isogenies of prime degree. (With an appendix by D. Goldfeld)
• The weight in Serre's conjectures on modular forms
• On modular representations of \(\text{Gal}(\overline{\mathbb Q}/\mathbb Q)\) arising from modular forms
• Modular elliptic curves and Fermat's Last Theorem
• Modularity of a family of elliptic curves
• Catalan's conjecture
• An application of the symplectic argument to some Fermat-type equations
• Conductors of $\ell $-adic representations
• The Problem of Catalan
• Algorithm for determining the type of a singular fiber in an elliptic pencil
• On the equation $a^p + 2^α b^p + c^p = 0$
• The Arithmetic of Elliptic Curves
• Majorations Effectives Pour L’ Équation de Fermat Généralisée
• Courbes de Fermat: résultats et problèmes
• Primary cyclotomic units and a proof of Catalans conjecture
• On the Equations z^m = F (x, y ) and Ax^p + By^q = Cz^r
• Number Theory
• Unnamed Item
• Unnamed Item
• Unnamed Item