Diophantine Analysis

by Steuding; Jorn

• ISBN13: 9781584884828
• ISBN10: 1584884827
• Edition: 1st
• Format: Hardcover
• Copyright: 2005-05-19
• Publisher: Chapman & Hall/

Summary

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 approximations and the theory of diophantine equations, with emphasis on interactions between these subjects. Beginning with the basic principles, the author develops his treatment around the theory of continued fractions and examines the classic theory, including some of its applications. He also explores modern topics rarely addressed in other texts, including the abc conjecture, the polynomial Pell equation, and the irrationality of the zeta function and touches on topics and applications related to discrete mathematics, such as factoring methods for large integers.Setting the stage for tackling the field's many open problems and conjectures, Diophantine Analysis is an ideal introduction to the fundamentals of this venerable but still dynamic field. A detailed appendix supplies the necessary background material, more than 200 exercises reinforce the concepts, and engaging historical notes bring the subject to life.

Author Biography

Jorn Steuding is a professor in the Department of Mathematics, Universidad Autonoma de Madrid, Spain.

Table of Contents

Preface
Introduction: basic principles
Who was Diophantus?	1	(1)
Pythagorean triples	2	(1)
Fermat's last theorem	3	(1)
The method of infinite descent	4	(2)
Cantor's paradise	6	(1)
Irrationality of e	7	(1)
Irrationality of π	8	(2)
Approximating with rationals	10	(2)
Linear diophantine equations	12	(5)
Exercises	14	(3)
Classical approximation theorems
Dirichlet's approximation theorem	17	(2)
A first irrationality criterion	19	(1)
The order of approximation	19	(2)
Kronecker's approximation theorem	21	(1)
Billiard	22	(1)
Uniform distribution	23	(2)
The Farey sequence	25	(1)
Mediants and Ford circles	26	(2)
Hurwitz' theorem	28	(2)
Pade approximation	30	(6)
Exercises	32	(4)
Continued fractions
The Euclidean algorithm revisited and calendars	36	(1)
Finite continued fractions	37	(2)
Interlude: Egyptian fractions	39	(3)
Infinite continued fractions	42	(1)
Approximating with convergents	43	(1)
The law of best approximations	44	(1)
Consecutive convergents	45	(1)
The continued fraction for e	46	(6)
Exercises	49	(3)
The irrationality of ζ(3)
The Riemann zeta-function	52	(2)
Apery's theorem	54	(1)
Approximating ζ(3)	54	(2)
A recursion formula	56	(2)
The speed of convergence	58	(2)
Final steps in the proof	60	(2)
An irrationality measure	62	(1)
A non-simple continued fraction	63	(1)
Beukers' proof	64	(7)
Notes on recent results	66	(1)
Exercises	66	(5)
Quadratic irrationals
Fibonacci numbers and paper folding	71	(2)
Periodic continued fractions	73	(2)
Galois' theorem	75	(2)
Square roots	77	(1)
Equivalent numbers	78	(1)
Serret's theorem	79	(1)
The Markoff spectrum	80	(2)
Badly approximable numbers	82	(6)
Notes on the metric theory	82	(2)
Exercises	84	(4)
The Pell equation
The cattle problem	88	(2)
Lattice points on hyperbolas	90	(2)
An infinitude of solutions	92	(2)
The minimal solution	94	(1)
The group of solutions	95	(1)
The minus equation	96	(1)
The polynomial Pell equation	97	(3)
Nathanson's theorem	100	(7)
Notes for further reading	102	(1)
Exercises	103	(4)
Factoring with continued fractions
The RSA cryptosystem	107	(2)
A diophantine attack on RSA	109	(1)
An old idea of Fermat	110	(2)
CFRAC	112	(3)
Examples of failures	115	(1)
Weighted mediants and a refinement	115	(5)
Notes on primality testing	117	(1)
Exercises	118	(2)
Geometry of numbers
Minkowski's convex body theorem	120	(2)
General lattices	122	(2)
The lattice basis theorem	124	(1)
Sums of squares	125	(3)
Applications to linear and quadratic forms	128	(1)
The shortest lattice vector problem	129	(2)
Gram-Schmidt and consequences	131	(1)
Lattice reduction in higher dimensions	132	(2)
The LLL-algorithm	134	(2)
The small integer problem	136	(5)
Notes on sphere packings	136	(1)
Exercises	137	(4)
Transcendental numbers
Algebraic vs. transcendental	141	(1)
Liouville's theorem	142	(2)
Liouville numbers	144	(1)
The transcendence of e	145	(2)
The transcendence of π	147	(2)
Squaring the circle?	149	(6)
Notes on transcendental numbers	151	(1)
Exercises	152	(3)
The theorem of Roth
Roth's theorem	155	(1)
Thue equations	156	(2)
Finite vs. infinite	158	(2)
Differential operators and indices	160	(2)
Outline of Roth's method	162	(2)
Siegel's lemma	164	(1)
The index theorem	165	(2)
Wronskians and Roth's lemma	167	(4)
Final steps in Roth's proof	171	(6)
Notes for further reading	173	(1)
Exercises	174	(3)
The abc-conjecture
Hilbert's tenth problem	177	(2)
The ABC-theorem for polynomials	179	(2)
Fermat's last theorem for polynomials	181	(1)
The polynomial Pell equation revisited	182	(1)
The abc-conjecture	183	(1)
LLL & abc	184	(2)
The Erdos-Woods conjecture	186	(1)
Fermat, Catalan & co.	187	(2)
Mordell's conjecture	189	(6)
Notes on abc	190	(2)
Exercises	192	(3)
p-adic numbers
Non-Archimedean valuations	195	(1)
Ultrametric topology	196	(2)
Ostrowski's theorem	198	(2)
Curious convergence	200	(1)
Characterizing rationals	201	(2)
Completions of the rationals	203	(2)
p-adic numbers as power series	205	(1)
Error-free computing	206	(7)
Notes on the p-adic interpolation of the zeta-function	207	(1)
Exercises	208	(5)
Hensel's lemma and applications
p-adic integers	213	(1)
Solving equations in p-adic numbers	214	(2)
Hensel's lemma	216	(2)
Units and squares	218	(1)
Roots of unity	219	(1)
Hensel's lemma revisited	220	(1)
Hensel lifting: factoring polynomials	221	(6)
Notes on p-adics: what we leave out	224	(1)
Exercises	224	(3)
The local-global principle
One for all and all for one	227	(1)
The theorem of Hasse-Minkowski	228	(1)
Ternary quadratics	229	(3)
The theorems of Chevalley and Warning	232	(2)
Applications and limitations	234	(2)
The local Fermat problem	236	(3)
Exercises	237	(2)
Appendix A. Algebra and number theory
A.1. Groups, rings, and fields	239	(2)
A.2. Prime numbers	241	(1)
A.3. Riemann's hypothesis	242	(1)
A.4. Modular arithmetic	243	(2)
A.5. Quadratic residues	245	(1)
A.6. Polynomials	246	(1)
A.7. Algebraic number fields	247	(2)
A.8. Kummer's work on Fermat's last theorem	249	(2)
Bibliography	251	(7)
Index	258