A Friendly Introduction to Number Theory

by Silverman, Joseph H.

• ISBN13: 9780321816191
• ISBN10: 0321816196
• Edition: 4th
• Format: Hardcover
• Copyright: 2012-01-18
• Publisher: Pearson

Summary

A Friendly Introduction to Number Theory, Fourth Editionis designed to introduce readers to the overall themes and methodology of mathematics through the detailed study of one particular facet-number theory. Starting with nothing more than basic high school algebra, readers are gradually led to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing is appropriate for the undergraduate audience and includes many numerical examples, which are analyzed for patterns and used to make conjectures. Emphasis is on the methods used for proving theorems rather than on specific results.

Author Biography

Joseph H. Silverman is a Professor of Mathematics at Brown University. He received his Sc.B. at Brown and his Ph.D. at Harvard, after which he held positions at MIT and Boston University before joining the Brown faculty in 1988. He has published more than 100 peer-reviewed research articles and seven books in the fields of number theory, elliptic curves, arithmetic geometry, arithmetic dynamical systems, and cryptography.  He is a highly regarded teacher, having won teaching awards from Brown University and the Mathematical Association of America, as well as a Steele Prize for Mathematical Exposition from the American Mathematical Society. He has supervised the theses of more than 25 Ph.D. students, is a co-founder of NTRU Cryptosystems, Inc., and has served as an elected member of the American Mathematical Society Council and Executive Committee.

Table of Contents

Preface	p. v
Flowchart of Chapter Dependencies	p. ix
Introduction	p. 1
What Is Number Theory?	p. 6
Pythagorean Triples	p. 13
Pythagorean Triples and the Unit Circle	p. 21
Sums of Higher Powers and Fermat's Last Theorem	p. 26
Divisibility and the Greatest Common Divisor	p. 30
Linear Equations and the Greatest Common Divisor	p. 37
Factorization and the Fundamental Theorem of Arithmetic	p. 46
Congruences	p. 55
Congruences, Powers, and Fermat's Little Theorem	p. 65
Congruences, Powers, and Euler's Formula	p. 71
Euler's Phi Function and the Chinese Remainder Theorem	p. 75
Prime Numbers	p. 83
Counting Primes	p. 90
Mersenne Primes	p. 96
Mersenne Primes and Perfect Numbers	p. 101
Powers Modulo m and Successive Squaring	p. 111
Computing kth Roots Modulo m	p. 118
Powers, Roots, and "Unbreakable" Codes	p. 123
Primality Testing and Carmichael Numbers	p. 129
Squares Modulo p	p. 141
Is-1 a Square Modulo p? Is 2?	p. 148
Quadratic Reciprocity	p. 159
Proof of Quadratic Reciprocity	p. 171
Which Primes Are Sums of Two Squares?	p. 181
Which Numbers Are Sums of Two Squares?	p. 193
As Easy as One, Two, Three	p. 199
Euler's Phi Function and Sums of Divisors	p. 206
Powers Modulo p and Primitive Roots	p. 211
Primitive Roots and Indices	p. 224
The Equation X4+Y4=Z4	p. 231
Square-Triangular Numbers Revisited	p. 236
Pell's Equation	p. 245
Diophantine Approximation	p. 251
Diophantine Approximation and Pell's Equation	p. 260
Number Theory and Imaginary Numbers	p. 267
The Gaussian Integers and Unique Factorization	p. 281
Irrational Numbers and Transcendental Numbers	p. 297
Binomial Coefficients and Pascal's Triangle	p. 313
Fibonacci's Rabbits and Linear Recurrence Sequences	p. 324
Oh, What a Beautiful Function	p. 339
Cubic Curves and Elliptic Curves	p. 353
Elliptic Curves with Few Rational Points	p. 366
Points on Elliptic Curves Modulo p	p. 373
Torsion Collections Modulo p and Bad Primes	p. 384
Defect Bounds and Modularity Patterns	p. 388
Elliptic Curves and Fermat's Last Theorem	p. 394
Further Reading	p. 396
Index	p. 397
The Topsy-Turvy World of Continued Fractions [online]	p. 410
Continued Fractions and Pell's Equation [online]	p. 426
Generating Functions [online]	p. 442
Sums of Powers [online]	p. 452
Factorization of Small Composite Integers [online]	p. 464
A List of Primes [online]	p. 466
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