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Let K be a number field. An element b ∈ K* being given, let Cb be the curve defined over K by the equation x4 + y4 = bz4. Let Cb(K) be the set of the K-rational points of Cb. This paper uses Dem'janenko and Manin type methods to obtain effective criteria for Cb(K) to be empty.
Let $A > 0$ be an integer. The equation $x^5 - y^5 = Az^5$ was first studied by Dirichlet and Lebesgue. Lebesgue conjectured in 1843 that if $A$ has no prime divisors of the form $10k+1$, the equation has no solutions except the visible ones. Partial results were obtained by Lebesgue and by Terjanian in 1987. The purpose of the paper is to prove Lebesgue's conjecture. The main tool used is the method known as the elliptic Chabauty method.
Let p be a prime number, K be a finite non-ramified extension of ℚp and E be an elliptic curve defined over K. Let K(Ep) be the field of the p-division points of E. In this paper we determine the valuation of the different of the extension K(Ep)/K.