Displaying similar documents to “Heights in number fields”

The Hooley-Huxley contour method for problems in number fields III : frobenian functions

Mark D. Coleman (2001)

Journal de théorie des nombres de Bordeaux

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In this paper we study finite valued multiplicative functions defined on ideals of a number field and whose values on the prime ideals depend only on the Frobenius class of the primes in some Galois extension. In particular we give asymptotic results when the ideals are restricted to “small regions”. Special cases concern Ramanujan's tau function in small intervals and relative norms in “small regions” of elements from a full module of the Galois extension.

Catalan’s conjecture

Yuri F. Bilu (2002-2003)

Séminaire Bourbaki

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The subject of the talk is the recent work of Mihăilescu, who proved that the equation x p - y q = 1 has no solutions in non-zero integers x , y and odd primes p , q . Together with the results of Lebesgue (1850) and Ko Chao (1865) this implies the celebrated: Before the work of Mihăilescu the most definitive result on Catalan’s problem was due to Tijdeman (1976), who proved that the solutions of Catalan’s equation are bounded by an absolute effective constant.