勷勤数学•领军学者报告

题      目：Exponential Diophantine equations

报  告  人： BUGEAUD YANN DANIEL FRANCOIS 教授 

                      (邀请人：袁平之)

                                      斯特拉斯堡大学（University of Strasbourg )

时      间： 3月27日  15：30-16:30

地     点：数科院东楼304

报告人简介:

        BUGEAUD YANN DANIEL FRANCOIS 法国斯特拉斯堡大学教授，法国大学研究院资深成员，国家著名数论专家。主要研究兴趣包括丢番图逼近与丢番图方程、模1分布和超越数论。Yann Bugeaud教授已在Ann of Math、IMRN等国际顶刊等刊物上发表学术论文250多篇，出版专著4本。目前是Acta Arithmetic等五个国际数论期刊的编委。

摘      要：

       A Diophantine equation is a polynomial equation with integer coefficients, for which only integer solutions are of interest. An exponential Diophantine equation is one in which unknowns can appear in exponents. Examples include the Fermat equation x^n + y^n = z^n and the Catalan equation x^m- y^n = 1. We will start with a presentation of Baker’s theory of linear forms in the logarithms of algebraic numbers and of some of its applications to get effective upper bounds for the solutions of classical families of Diophantine equations. But Baker’s theory also has striking applications to exponential Diophantine equations: it allowed Tijdeman to prove in 1976 that the Catalan equation has only finitely many solutions. More recently, in 2006, Bugeaud, Mignotte, and Siksek established that 1, 8, and 144 are the only perfect powers in the Fibonacci sequence. We will conclude by  mentioning several other Diophantine questions involving perfect powers, sets of integers whose b-ary expansions has few digits, and sets of integers composed by only finitely many (given) primes.

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