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.

Full Product Details

Author:   W.S. Anglin
Publisher:   Springer
Imprint:   Springer
Edition:   Softcover reprint of the original 1st ed. 1995
Volume:   8
Dimensions:   Width: 16.00cm , Height: 2.10cm , Length: 24.00cm
Weight:   0.643kg
ISBN:  

9789401041263

ISBN 10:   9401041261
Pages:   390
Publication Date:   21 December 2011
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand  
We will order this item for you from a manufactured on demand supplier.

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.

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