A Friendly Introduction to Number Theory

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  • Edition: 4th
  • Format: Hardcover
  • Copyright: 2012-01-18
  • Publisher: Pearson
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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

Prefacep. v
Flowchart of Chapter Dependenciesp. ix
Introductionp. 1
What Is Number Theory?p. 6
Pythagorean Triplesp. 13
Pythagorean Triples and the Unit Circlep. 21
Sums of Higher Powers and Fermat's Last Theoremp. 26
Divisibility and the Greatest Common Divisorp. 30
Linear Equations and the Greatest Common Divisorp. 37
Factorization and the Fundamental Theorem of Arithmeticp. 46
Congruencesp. 55
Congruences, Powers, and Fermat's Little Theoremp. 65
Congruences, Powers, and Euler's Formulap. 71
Euler's Phi Function and the Chinese Remainder Theoremp. 75
Prime Numbersp. 83
Counting Primesp. 90
Mersenne Primesp. 96
Mersenne Primes and Perfect Numbersp. 101
Powers Modulo m and Successive Squaringp. 111
Computing kth Roots Modulo mp. 118
Powers, Roots, and "Unbreakable" Codesp. 123
Primality Testing and Carmichael Numbersp. 129
Squares Modulo pp. 141
Is-1 a Square Modulo p? Is 2?p. 148
Quadratic Reciprocityp. 159
Proof of Quadratic Reciprocityp. 171
Which Primes Are Sums of Two Squares?p. 181
Which Numbers Are Sums of Two Squares?p. 193
As Easy as One, Two, Threep. 199
Euler's Phi Function and Sums of Divisorsp. 206
Powers Modulo p and Primitive Rootsp. 211
Primitive Roots and Indicesp. 224
The Equation X4+Y4=Z4p. 231
Square-Triangular Numbers Revisitedp. 236
Pell's Equationp. 245
Diophantine Approximationp. 251
Diophantine Approximation and Pell's Equationp. 260
Number Theory and Imaginary Numbersp. 267
The Gaussian Integers and Unique Factorizationp. 281
Irrational Numbers and Transcendental Numbersp. 297
Binomial Coefficients and Pascal's Trianglep. 313
Fibonacci's Rabbits and Linear Recurrence Sequencesp. 324
Oh, What a Beautiful Functionp. 339
Cubic Curves and Elliptic Curvesp. 353
Elliptic Curves with Few Rational Pointsp. 366
Points on Elliptic Curves Modulo pp. 373
Torsion Collections Modulo p and Bad Primesp. 384
Defect Bounds and Modularity Patternsp. 388
Elliptic Curves and Fermat's Last Theoremp. 394
Further Readingp. 396
Indexp. 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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