Effective Methods for Diophantine Equations
Effective Methods for Diophantine Equations
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丢番图方程的有效方法
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发表时间：
2005
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通讯作者：
Szabolcs Tengely
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作者：
Szabolcs Tengely
Proefschrift ter verkrijging货车de graad货车博士aan de Universiteit Leiden，op gezag货车de Rector Magnificus博士目录1介绍1 2龙格型丢番图方程11 2.参考书目65 Samenvatting 73简历74 vii本论文包含以下论文的材料。引言在本文中，我们将有效地解决丢番图方程的各种方法，更准确地说，由龙格的方法，贝克的方法和Chabauty的方法。为了把我们的结果放在适当的背景下，我们总结了一些相关的历史。丢番图方程是f（x 1，x 2，. ..，xn）= 0，其中f是给定的函数，并且未知数x1，x2，. ..，xn必须是有理数或整数。作为概念的推广，人们可以考虑在数域上的有理解或积分解。在丢番图方程的研究中，有一些自然的问题：·方程可解吗？·解的个数是有限的还是无限的？·是否可以确定所有的解决方案？丢番图是一位数学家，他生活在公元300年左右的亚历山大。丢番图的《算术》中有13本书，其中有6本是希腊语的。最著名的拉丁文翻译是由于Bachet在1621年。1968年，在伊朗发现了一份阿拉伯语手稿，这是一份在亚历山大写的希腊文本的翻译，但很可能是由丢番图的一些评论员写的。在他的作品中，他指出数学问题，并提供合理的解决方案。为了让我们对这里提到的问题有一个概念，其中有两个问题。第一个问题是（第4册的问题20）找到四个数字，使得其中任何两个数字的乘积加上1是一个完美的平方。具有此属性的集合称为（有理）丢番图四元组。第二个问题是问题17册6的阿拉伯手稿算术归结为寻找积极的合理解决方案y 2 = x 6 + x 2 + 1。丢番图构造了解x = 1 2，y = 9 8。费马大定理是关于丢番图方程xn + yn = z n的。费马（1601-1665）在丢番图的一本书的空白处写道：
Proefschrift ter verkrijging van de graad van Doctor aan de Universiteit Leiden, op gezag van de Rector Magnificus Dr. Contents 1 Introduction 1 2 Runge-type Diophantine Equations 11 2. Bibliography 65 Samenvatting 73 Curriculum Vitae 74 vii This thesis contains material from the following papers. Introduction In the thesis we shall solve Diophantine equations effectively by various methods, more precisely by Runge's method, Baker's method and Chabauty's method. To put our results in the proper context we summarize some of the relevant history. A Diophantine equation is an equation of the form f (x 1 , x 2 ,. .. , x n) = 0, where f is a given function and the unknowns x 1 , x 2 ,. .. , x n are required to be rational numbers or to be integers. As a generalisation of the concept one may consider rational or integral solutions over a number field. In the study of Diophantine equations there are some natural questions: • Is the equation solvable? • Is the number of solutions finite or infinite? • Is it possible to determine all solutions? Diophantus was a mathematician who lived in Alexandria around 300 A.D. Six Greek books out of thirteen of Diophantus' Arithmetica have been known for a long time. The most famous Latin translation is due to Bachet in 1621. In 1968 an Arabic manuscript was found in Iran, which is a translation from a Greek text written in Alexandria, but probable it was written by some of Diophantus' commentators. In his works he stated mathematical problems and provided rational solutions. To give an idea of the kind of problems we mention here two of them. The first problem is (problem 20 of book 4) to find four numbers such that the product of any two of them increased by 1 is a perfect square. A set with this property is called a (rational) Diophantine quadruple. The set with this property which Diophantus constructed The second problem is problem 17 of book 6 of the Arabic manuscript of Arithmetica which comes down to find positive rational solutions to y 2 = x 6 + x 2 + 1. Diophantus constructed the solution x = 1 2 , y = 9 8. Fermat's Last Theorem concerns the Diophantine equation x n + y n = z n. Fermat (1601-1665) wrote in the margin of an edition of Diophantus' book …