Diophantine Equations And Inequalities In Algebraic Number Fields
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Diophantine Equations and Inequalities in Algebraic Number Fields
Author | : Yuan Wang |
Publsiher | : Springer Science & Business Media |
Total Pages | : 185 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 9783642581717 |
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The circle method has its genesis in a paper of Hardy and Ramanujan (see [Hardy 1])in 1918concernedwiththepartitionfunction andtheproblemofrep resenting numbers as sums ofsquares. Later, in a series of papers beginning in 1920entitled "some problems of'partitio numerorum''', Hardy and Littlewood (see [Hardy 1]) created and developed systematically a new analytic method, the circle method in additive number theory. The most famous problems in ad ditive number theory, namely Waring's problem and Goldbach's problem, are treated in their papers. The circle method is also called the Hardy-Littlewood method. Waring's problem may be described as follows: For every integer k 2 2, there is a number s= s(k) such that every positive integer N is representable as (1) where Xi arenon-negative integers. This assertion wasfirst proved by Hilbert [1] in 1909. Using their powerful circle method, Hardy and Littlewood obtained a deeper result on Waring's problem. They established an asymptotic formula for rs(N), the number of representations of N in the form (1), namely k 1 provided that 8 2 (k - 2)2 - +5. Here
Discriminant Equations in Diophantine Number Theory
Author | : Jan-Hendrik Evertse,Klmn Gyory |
Publsiher | : Cambridge University Press |
Total Pages | : 477 |
Release | : 2016-11-03 |
Genre | : Mathematics |
ISBN | : 9781107097612 |
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The first comprehensive and up-to-date account of discriminant equations and their applications. For graduate students and researchers.
Unit Equations in Diophantine Number Theory
Author | : Jan-Hendrik Evertse,Klmn Gyory |
Publsiher | : Cambridge University Press |
Total Pages | : 381 |
Release | : 2015-12-30 |
Genre | : Mathematics |
ISBN | : 9781107097605 |
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A comprehensive, graduate-level treatment of unit equations and their various applications.
Number Theory
Author | : Daniel Duverney |
Publsiher | : World Scientific |
Total Pages | : 348 |
Release | : 2010 |
Genre | : Mathematics |
ISBN | : 9789814307451 |
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This textbook presents an elementary introduction to number theory and its different aspects: approximation of real numbers, irrationality and transcendence problems, continued fractions, diophantine equations, quadratic forms, arithmetical functions and algebraic number theory. These topics are covered in 12 chapters and more than 200 solved exercises. Clear, concise, and self-contained, this textbook may be used by undergraduate and graduate students, as well as highschool mathematics teachers. More generally, it will be suitable for all those who are interested in number theory, this fascinating branch of mathematics.
Equations and Inequalities
Author | : Jiri Herman,Radan Kucera,Jaromir Simsa |
Publsiher | : Springer Science & Business Media |
Total Pages | : 362 |
Release | : 2000-03-23 |
Genre | : Education |
ISBN | : 0387989420 |
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A look at solving problems in three areas of classical elementary mathematics: equations and systems of equations of various kinds, algebraic inequalities, and elementary number theory, in particular divisibility and diophantine equations. In each topic, brief theoretical discussions are followed by carefully worked out examples of increasing difficulty, and by exercises which range from routine to rather more challenging problems. While it emphasizes some methods that are not usually covered in beginning university courses, the book nevertheless teaches techniques and skills which are useful beyond the specific topics covered here. With approximately 330 examples and 760 exercises.
Quadratic Number Fields
Author | : Franz Lemmermeyer |
Publsiher | : Springer Nature |
Total Pages | : 348 |
Release | : 2021-09-18 |
Genre | : Mathematics |
ISBN | : 9783030786526 |
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This undergraduate textbook provides an elegant introduction to the arithmetic of quadratic number fields, including many topics not usually covered in books at this level. Quadratic fields offer an introduction to algebraic number theory and some of its central objects: rings of integers, the unit group, ideals and the ideal class group. This textbook provides solid grounding for further study by placing the subject within the greater context of modern algebraic number theory. Going beyond what is usually covered at this level, the book introduces the notion of modularity in the context of quadratic reciprocity, explores the close links between number theory and geometry via Pell conics, and presents applications to Diophantine equations such as the Fermat and Catalan equations as well as elliptic curves. Throughout, the book contains extensive historical comments, numerous exercises (with solutions), and pointers to further study. Assuming a moderate background in elementary number theory and abstract algebra, Quadratic Number Fields offers an engaging first course in algebraic number theory, suitable for upper undergraduate students.
A Brief Guide to Algebraic Number Theory
Author | : H. P. F. Swinnerton-Dyer |
Publsiher | : Cambridge University Press |
Total Pages | : 164 |
Release | : 2001-02-22 |
Genre | : Mathematics |
ISBN | : 0521004233 |
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Broad graduate-level account of Algebraic Number Theory, first published in 2001, including exercises, by a world-renowned author.
Diophantus and Diophantine Equations
Author | : Izabella Grigorʹevna Bashmakova,Joseph H. Silverman |
Publsiher | : Cambridge University Press |
Total Pages | : 110 |
Release | : 1997 |
Genre | : Mathematics |
ISBN | : 0883855267 |
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Semi-popular maths on an area of number theory related to Fermat.