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Local solubility of generalised Fermat equations
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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.
Forward citations
Cited by 2 Pith papers
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An improved large sieve for quadratic characters via Hooley neutralisers and its applications
Using Hooley neutralisers, a large sieve for quadratic characters is improved for multiplicatively weighted sequences, with applications to hyperbolic-region character sums.
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Solubility of a family of conics with polynomial coefficients in many variables
An asymptotic for the density of soluble conics with polynomial coefficients is proposed, but the sign decomposition (6.5) that bridges the circle-method counts to the true count is false.
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