Atnaujinkite slapukų nuostatas

Quadratic Diophantine Equations 2015 ed. [Kietas viršelis]

4.80/5 (10 ratings by Goodreads)
,
  • Formatas: Hardback, 211 pages, aukštis x plotis: 235x155 mm, weight: 4734 g, XVIII, 211 p., 1 Hardback
  • Serija: Developments in Mathematics 40
  • Išleidimo metai: 30-Jun-2015
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 0387351566
  • ISBN-13: 9780387351568
Kitos knygos pagal šią temą:
Quadratic Diophantine Equations 2015 ed.Didesnis paveikslėlis
  • Formatas: Hardback, 211 pages, aukštis x plotis: 235x155 mm, weight: 4734 g, XVIII, 211 p., 1 Hardback
  • Serija: Developments in Mathematics 40
  • Išleidimo metai: 30-Jun-2015
  • Leidėjas: Springer-Verlag New York Inc.
  • ISBN-10: 0387351566
  • ISBN-13: 9780387351568
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9780387351568.html
Raktažodžiai:
This monograph 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. These new techniques combined with the latest increases in computational power shed new light on important open problems. The authors motivate the study of quadratic Diophantine equations with excellent examples, open problems, and applications. Moreover, the exposition aptly demonstrates many applications of results and techniques from the study of Pell-type equations to other problems in number theory

The book is intended for advanced undergraduate and graduate students as well as researchers. It challenges the reader to apply not only specific techniques and strategies, but also to employ methods and tools from other areas of mathematics, such as algebra and analysis

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.



This book reveiws the last two decades of computational techniques and progress in the classical theory of quadratic diophantine equations. Presents important quadratic diophantine equations and applications, and includes excellent examples and open problems.

Recenzijos

The book under review is an excellent book on the interesting subject of quadratic Diophantine equations. It is well written, well organized, and contains a wealth of material that one does not expect to find in a book of its size, with full proofs of scores of theorems. This reviewer does not know any book that covers similar material, and sees it as a very valuable and much needed addition to the literature on number theory. (Mowaffaq Hajja, zbMATH 1376.11001, 2018)

Diophantine analysis aims to solve equations (usually polynomial) in integers (or rationals). this work settles the classical foundation, then develops state-of-the-art issues, especially concerning computation. Summing Up: Recommended. Lower-division undergraduates through professionals/practitioners. (D. V. Feldman, Choice, Vol. 53 (9), May, 2016)

The primary focus of this book under review is the integer solutions of Pell equations, their generalisations and related diophantine equations, along with applications of these equations. The book is suitable for readers from the level of a motivated undergraduate upwards, who are interested in the classical techniques for solving such equations. There is also an extensive bibliography. (Paul M. Voutier, Mathematical Reviews, March, 2016)

1 Why Quadratic Diophantine Equations?
1(8)
1.1 Thue's Theorem
1(1)
1.2 Hilbert's Tenth Problem
2(2)
1.3 Euler's Concordant Forms
4(1)
1.4 Trace of Hecke Operators for Maass Forms
5(1)
1.5 Diophantine Approximation and Numerical Integration
5(1)
1.6 Threshold Phenomena in Random Lattices and Reduction Algorithms
6(1)
1.7 Standard Homogeneous Einstein Manifolds and Diophantine Equations
6(1)
1.8 Computing Self-Intersections of Closed Geodesies
7(1)
1.9 Hecke Groups and Continued Fractions
7(1)
1.10 Sets of Type (m, n) in Projective Planes
8(1)
2 Continued Fractions, Diophantine Approximation, and Quadratic Rings
9(22)
2.1 Simple Continued Fractions
10(17)
2.1.1 The Euclidean Algorithm
10(2)
2.1.2 Uniqueness
12(1)
2.1.3 Infinite Continued Fractions
13(3)
2.1.4 Irrational Numbers
16(1)
2.1.5 Approximations to Irrational Numbers
17(2)
2.1.6 Best Possible Approximations
19(2)
2.1.7 Periodic Continued Fractions
21(6)
2.2 Units and Norms in Quadratic Rings
27(4)
2.2.1 Quadratic Rings
27(1)
2.2.2 Norms in Quadratic Rings
27(4)
3 Pell's Equation
31(24)
3.1 History and Motivation
31(2)
3.2 The General Solution by Elementary Methods
33(4)
3.3 The General Solution by Continued Fractions
37(3)
3.4 The General Solution by Quadratic Rings
40(4)
3.5 The Equation ax2 - by2 = 1
44(2)
3.6 The Negative Pell Equation and the Pell--Stevenhagen Constants
46(9)
4 General Pell's Equation
55(52)
4.1 General Theory
55(6)
4.2 Solvability of General Pell's Equation
61(8)
4.2.1 PDP and the Square Polynomial Problem
62(1)
4.2.2 The Legendre Test
63(1)
4.2.3 Legendre Unsolvability Tests
64(3)
4.2.4 Modulo n Unsolvability Tests
67(1)
4.2.5 Extended Multiplication Principle
68(1)
4.3 An Algorithm for Determining the Fundamental Solutions Based on Simple Continued Fractions (The LMM Method)
69(5)
4.3.1 An Algorithm for Solving the General Pell's Equation (4.1.1)
71(3)
4.4 Solving the General Pell's Equation
74(10)
4.4.1 The PQa Algorithm for Solving Pell's and Negative Pell's Equations
74(1)
4.4.2 Solving the Special Equations x2 - Dy2 = ±4
75(3)
4.4.3 Structure of Solutions to the General Pell's Equation
78(2)
4.4.4 Solving the Equation x2 - Dy2 = N for N < √D
80(1)
4.4.5 Solving the Equation x2 - Dy2 = N by Brute-Force Search
80(1)
4.4.6 Numerical Examples
80(4)
4.5 Solvability and Unsolvability of the Equation ax2 - by2 = c
84(5)
4.6 Solving the General Pell Equation by Using Quadratic Rings
89(1)
4.7 Another Algorithm for Solving General Pell's Equation
90(3)
4.8 The Diophantine Equation ax2 + bxy + cy2 = N
93(2)
4.9 Thue's Theorem and the Equations x2 - Dy2 = ±N
95(12)
4.9.1 Euclid's Algorithm and Thue's Theorem
96(1)
4.9.2 The Equation x2 - Dy2 = N with Small D
96(1)
4.9.3 The Equations x2 - 2y2 = ±N
97(2)
4.9.4 The Equations x2 - 3y2 = ±N
99(1)
4.9.5 The Equations x2 - 5y2 = ±N
100(2)
4.9.6 The Equations x2 - 6y2 = ±N
102(2)
4.9.7 The Equations x2 - 7y2 = ±N
104(3)
5 Equations Reducible to Pell's Type Equations
107(38)
5.1 The Equations x2 - kxy2 + y4 = 1 and x2 - kxy2 + y4 = 4
107(4)
5.2 The Equation x2n - Dy2 = 1
111(7)
5.3 The Equation x2 + (x + 1)2 + ... + (x + n - 1)2 = y2 + (y + 1)2 + ... + (y + n + k - 1)2
118(6)
5.4 The Equation x2 + 2(x + 1)2 + ... + n(x + n - 1)2 = y2
124(2)
5.5 The Equation (x2 + a)(y2 + b) = F2(x, y, z)
126(4)
5.5.1 The Equation x2 + y2 + z2 + 2xyz = 1
127(1)
5.5.2 The Equation x2 + y2 + z2 - xyz = 4
128(1)
5.5.3 The Equation (x2 + 1)(y2 + 1) = z2
129(1)
5.5.4 The Equation (x2 - 1)(y2 - 1) = (z2 - 1)2
129(1)
5.6 Other Equations with Infinitely Many Solutions
130(15)
5.6.1 The Equation x2 + axy + y2 = 1
130(2)
5.6.2 The Equation x2 + 1/y2 + 1 = a2 + 1
132(3)
5.6.3 The Equation (x + y + z)2 = xyz
135(2)
5.6.4 The Equation (x + y + z + t)2 = xyzt
137(3)
5.6.5 The Equation (x + y + z + t)2 = xyzt + 1
140(1)
5.6.6 The Equation x3 + y3 + r3 + t3 = n
141(4)
6 Diophantine Representations of Some Sequences
145(24)
6.1 Diophantine r-Representable Sequences
146(2)
6.2 A Property of Some Special Sequences
148(2)
6.3 The Equations x2 + axy + y2 = ±1
150(3)
6.4 Diophantine Representations of the Sequences Fibonacci, Lucas, and Pell
153(7)
6.5 Diophantine Representations of Generalized Lucas Sequences
160(9)
7 Other Applications
169(32)
7.1 When Are an + b and cn + d Simultaneously Perfect Squares?
169(2)
7.2 Triangular Numbers
171(7)
7.2.1 Triangular Numbers with Special Properties
171(5)
7.2.2 Rational Numbers Representable as Tm/Tn
176(1)
7.2.3 When Is Tm/Tn a Perfect Square?
177(1)
7.3 Polygonal Numbers
178(4)
7.4 Powerful Numbers
182(6)
7.4.1 The Density of Powerful Numbers
183(2)
7.4.2 Consecutive Powerful Numbers
185(2)
7.4.3 Gaps Between Powerful Numbers
187(1)
7.5 The Diophantine Face of a Problem Involving Matrices in M2(Z)
188(11)
7.5.1 Nil-Clean Matrices in M2(Z)
188(2)
7.5.2 The General Case
190(4)
7.5.3 The Example
194(1)
7.5.4 How the Example Was Found
195(4)
7.6 A Related Question
199(2)
References 201(8)
Index 209