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arXiv · 2501.17619

Local solubility of generalised Fermat equations

Peter Koymans·Ross Paterson·Tim Santens·Alec Shute
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Abstract

For every $n \geq 2$ we determine the asymptotic formula for the number of integer triples $(a,b,c)$ of bounded absolute value such that the generalised Fermat equation given by $ax^n+by^n+cz^n=0$ is everywhere locally soluble. We compute the leading constant, answering a question of Loughran--Rome--Sofos, and determine that the conjectures of Loughran--Smeets and Loughran--Rome--Sofos hold for such equations.

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Peter Koymans, Ross Paterson, Tim Santens, Alec Shute. 2025-01-29. Local solubility of generalised Fermat equations. https://arxiv.org/abs/2501.17619

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