The Queen of Mathematics: An Introduction to Number Theory

by W.S. Anglin

Overview

Like other introductions to number theory, this one includes the usual curtsy to divisibility theory, the bow to congruence, and the little chat with quadratic reciprocity. It also includes proofs of results such as Lagrange's Four Square Theorem, the theorem behind Lucas's test for perfect numbers, the theorem that a regular n-gon is constructible just in case phi(n) is a power of 2, the fact that the circle cannot be squared, Dirichlet's theorem on primes in arithmetic progressions, the Prime Number Theorem, and Rademacher's partition theorem.
We have made the proofs of these theorems as elementary as possible.
Unique to The Queen of Mathematics are its presentations of the topic of palindromic simple continued fractions, an elementary solution of Lucas's square pyramid problem, Baker's solution for simultaneous Fermat equations, an elementary proof of Fermat's polygonal number conjecture, and the Lambek-Moser-Wild theorem.

Product Details

ISBN-13:	9789401041263
Publisher:	Springer Netherlands
Publication date:	12/21/2011
Series:	Texts in the Mathematical Sciences , #8
Edition description:	Softcover reprint of the original 1st ed. 1995
Pages:	390
Product dimensions:	6.30(w) x 9.45(h) x 0.03(d)

Table of Contents

1 Propaedeutics.- 1.1 Mathematical Induction.- 1.2 Bernoulli Numbers.- 1.3 Primes.- 1.4 Perfect Numbers.- 1.5 Greatest Integer function.- 1.6 Pythagorean Triangles.- 1.7 Diophantine Equations.- 1.8 Four Square Theorem.- 1.9 Fermat’s Last Theorem.- 1.10 Congruent Numbers.- 1.11 Möbius function.- 2 Simple Continued Fractions.- 2.1 Convergents and Convergence.- 2.2 Uniqueness of SCF Expansions.- 2.3 SCF Expansions of Rationals.- 2.4 Farey Series.- 2.5 Ax + By = C.- 2.6 SCF Approximations.- 2.7 SCF Expansions of Quadratic Surds.- 2.8 Periodic SCF Expansions.- 2.9 Pell Equation.- 2.10 Prefaced Palindromes.- 3 Congruence.- 3.1 Basic Properties.- 3.2 Euler’s—-Function.- 3.3 Primitive Roots.- 3.4 Decimal Expansions.- 3.5 x2— R (mod C).- 3.6 Palindromic SCF’s.- 3.7 Sums of Two Squares.- 3.8 Quadratic Residues.- 3.9 Theorema Aureum.- 3.10 Jacobi Symbol.- 3.11 More on x2— R (mod C).- 3.12 Ax2 + By = C.- 4 x2?Ry2 = C.- 4.1 SCF Solution.- 4.2 Recursive Formulas for Solutions.- 4.3 Ax2 + Bxy + Cy2 + Dx + Ey = F.- 4.4 Square Pyramid Problem.- 4.5 Lucas’s Test for Perfect Numbers.- 4.6 Simultaneous Fermat Equations.- 5 Classical Construction Problems.- 5.1 Euclidean Constructions.- 5.2 Fields and Vector Spaces.- 5.3 Limits of Ruler and Compass Construction.- 5.4 Gauss’s Constructions.- 5.5 Fermat Primes.- 5.6 The Transcendence of—.- 6 The Polygonal Number Theorem.- 6.1 Gaussian Forms.- 6.2 Ternary Quadratic Form Matrices.- 6.3 Omega Kernel or Square Forms.- 6.4 Ambiguous or Self-Inverse Forms.- 6.5 Sums of Triangular Numbers.- 6.6 Cauchy’s Proof.- 7 Analytic Number Theory.- 7.1 Characters.- 7.2 Dirichlet Series.- 7.3 Mangoldt function.- 7.4 L(1,X)— 0.- 7.5 Dirichlet’s Theorem on Primes in AP.- 7.6 How Many Pythagorean Triangles?.- 7.7 Prime Preliminaries.-7.8 Prime Number Theorem Proof.- 7.9 Partitions.- 7.10 Euler’s Power Series.- 7.11 A Fractal Path of Ford Circles.- 7.12 Möbius Transformations.- 7.13 Dedekind Sums.- 7.14 Eta function.- 7.15 Bessel Functions Avoided.- 7.16 Rademacher’s Proof.- 7.17 Numerical Calculations.- A Appendix: Answers to Selected Exercises.

Editorial Reviews

A textbook for first-year graduate or honors undergraduate students of mathematics. The only prerequisite for most of the book is pre-calculus mathematics, but many of the problems are not accessible to someone without several undergraduate mathematics course. Includes the standard topics of introductions to number theory, including divisibility theory, congruence, and quadratic reciprocity. Also proves that the circle cannot be squared. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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