Author : Jorn Steuding
Publisher : CRC Press
Page : 271 pages
File Size : 48,43 MB
Release : 2005-05-19
Category : Mathematics
ISBN : 1420057200
While its roots reach back to the third century, diophantine analysis continues to be an extremely active and powerful area of number theory. Many diophantine problems have simple formulations, they can be extremely difficult to attack, and many open problems and conjectures remain. Diophantine Analysis examines the theory of diophantine ap
Author : Robert Daniel Carmichael
Publisher :
Page : 138 pages
File Size : 27,71 MB
Release : 1915
Category : Diophantine analysis
ISBN :
Author : Umberto Zannier
Publisher : Springer
Page : 248 pages
File Size : 36,51 MB
Release : 2015-05-05
Category : Mathematics
ISBN : 8876425179
These lecture notes originate from a course delivered at the Scuola Normale in Pisa in 2006. Generally speaking, the prerequisites do not go beyond basic mathematical material and are accessible to many undergraduates. The contents mainly concern diophantine problems on affine curves, in practice describing the integer solutions of equations in two variables. This case historically suggested some major ideas for more general problems. Starting with linear and quadratic equations, the important connections with Diophantine Approximation are presented and Thue's celebrated results are proved in full detail. In later chapters more modern issues on heights of algebraic points are dealt with, and applied to a sharp quantitative treatment of the unit equation. The book also contains several supplements, hinted exercises and an appendix on recent work on heights.
Author : S. Lang
Publisher : Springer Science & Business Media
Page : 270 pages
File Size : 49,37 MB
Release : 2013-06-29
Category : Mathematics
ISBN : 3662070103
It is possible to write endlessly on elliptic curves. (This is not a threat.) We deal here with diophantine problems, and we lay the foundations, especially for the theory of integral points. We review briefly the analytic theory of the Weierstrass function, and then deal with the arithmetic aspects of the addition formula, over complete fields and over number fields, giving rise to the theory of the height and its quadraticity. We apply this to integral points, covering the inequalities of diophantine approximation both on the multiplicative group and on the elliptic curve directly. Thus the book splits naturally in two parts. The first part deals with the ordinary arithmetic of the elliptic curve: The transcendental parametrization, the p-adic parametrization, points of finite order and the group of rational points, and the reduction of certain diophantine problems by the theory of heights to diophantine inequalities involving logarithms. The second part deals with the proofs of selected inequalities, at least strong enough to obtain the finiteness of integral points.
Author : Edward B. Burger
Publisher : American Mathematical Soc.
Page : 160 pages
File Size : 15,96 MB
Release : 2000
Category : Mathematics
ISBN : 0821826409
The minimal background requirements and the author's fresh approach make this book enjoyable and accessible to a wide range of students, mathematicians, and fans of number theory."--BOOK JACKET.
Author : Titu Andreescu
Publisher : Springer Science & Business Media
Page : 350 pages
File Size : 17,59 MB
Release : 2010-09-02
Category : Mathematics
ISBN : 0817645497
This problem-solving book is an introduction to the study of Diophantine equations, a class of equations in which only integer solutions are allowed. The presentation features some classical Diophantine equations, including linear, Pythagorean, and some higher degree equations, as well as exponential Diophantine equations. Many of the selected exercises and problems are original or are presented with original solutions. An Introduction to Diophantine Equations: A Problem-Based Approach is intended for undergraduates, advanced high school students and teachers, mathematical contest participants — including Olympiad and Putnam competitors — as well as readers interested in essential mathematics. The work uniquely presents unconventional and non-routine examples, ideas, and techniques.
Author : Henri Cohen
Publisher : Springer Science & Business Media
Page : 673 pages
File Size : 34,44 MB
Release : 2008-10-10
Category : Mathematics
ISBN : 0387499237
The central theme of this book is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects. The book contains more than 350 exercises and the text is largely self-contained. Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included five appendices on these techniques.
Author : Isabella Grigoryevna Bashmakova
Publisher : American Mathematical Soc.
Page : 106 pages
File Size : 47,41 MB
Release : 2019-01-18
Category : Mathematics
ISBN : 1470450488
This book tells the story of Diophantine analysis, a subject that, owing to its thematic proximity to algebraic geometry, became fashionable in the last half century and has remained so ever since. This new treatment of the methods of Diophantus—a person whose very existence has long been doubted by most historians of mathematics—will be accessible to readers who have taken some university mathematics. It includes the elementary facts of algebraic geometry indispensable for its understanding. The heart of the book is a fascinating account of the development of Diophantine methods during the Renaissance and in the work of Fermat. This account is continued to our own day and ends with an afterword by Joseph Silverman, who notes the most recent developments including the proof of Fermat's Last Theorem.
Author : Titu Andreescu
Publisher : Springer
Page : 224 pages
File Size : 31,74 MB
Release : 2015-06-29
Category : Mathematics
ISBN : 0387541098
This text treats the classical theory of quadratic diophantine equations and guides the reader through the last two decades of computational techniques and progress in the area. The presentation features two basic methods to investigate and motivate the study of quadratic diophantine equations: the theories of continued fractions and quadratic fields. It also discusses Pell’s equation and its generalizations, and presents some important quadratic diophantine equations and applications. The inclusion of examples makes this book useful for both research and classroom settings.
Author : S. Lang
Publisher : Springer Science & Business Media
Page : 383 pages
File Size : 35,88 MB
Release : 2013-06-29
Category : Mathematics
ISBN : 1475718101
Diophantine problems represent some of the strongest aesthetic attractions to algebraic geometry. They consist in giving criteria for the existence of solutions of algebraic equations in rings and fields, and eventually for the number of such solutions. The fundamental ring of interest is the ring of ordinary integers Z, and the fundamental field of interest is the field Q of rational numbers. One discovers rapidly that to have all the technical freedom needed in handling general problems, one must consider rings and fields of finite type over the integers and rationals. Furthermore, one is led to consider also finite fields, p-adic fields (including the real and complex numbers) as representing a localization of the problems under consideration. We shall deal with global problems, all of which will be of a qualitative nature. On the one hand we have curves defined over say the rational numbers. Ifthe curve is affine one may ask for its points in Z, and thanks to Siegel, one can classify all curves which have infinitely many integral points. This problem is treated in Chapter VII. One may ask also for those which have infinitely many rational points, and for this, there is only Mordell's conjecture that if the genus is :;;; 2, then there is only a finite number of rational points.
