Friendly Introduction to Number Theory

Friendly Introduction to Number Theory

by Joseph H. Silverman


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Friendly Introduction to Number Theory by Joseph H. Silverman

This introductory text is designed to entice non-math focused individuals into learning some mathematics, while teaching them to think mathematically. Starting with nothing more than basic high school algebra, the reader is gradually led from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing style is informal and includes many numerical examples, which are analyzed for patterns and used to make conjectures. The emphasis is on the methods used for proving theorems rather than on specific results. Pythagorean Triples, Linear Equations and the Greatest Common Divisor, Factorization and the Fundamental Theorem of Arithmetic, Congruences, Mersenne Primes, Squares Modulo p, Quadratic Reciprocity, Pell's Equation, Diophantine Approximation, Irrational Numbers and Transcendental Numbers, Sums of Powers, Binomial Coefficients and Pascal's Triangle, Elliptic Curves and Fermat's Last Theorem. For individuals with limited math experience who are interested in number theory.

Product Details

ISBN-13: 9780131984523
Publisher: Pearson Educacion
Publication date: 08/28/2005

About the Author

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

1. What Is Number Theory?
2. Pythagorean Triples.
3. Pythagorean Triples and the Unit Circle.
4. Sums of Higher Powers and Fermat's Last Theorem.
5. Divisibility and the Greatest Common Divisor.
6. Linear Equations and the Greatest Common Divisor.
7. Factorization and the Fundamental Theorem of Arithmetic.
8. Congruences.
9. Congruences, Powers, and Fermat's Little Theorem.
10. Congruences, Powers, and Euler's Formula.
11. Euler's Phi Function.
12. Prime Numbers.
13. Counting Primes.
14. Mersenne Primes.
15. Mersenne Primes and Perfect Numbers.
16. Powers Modulo m and Successive Squaring.
17. Computing kth Roots Modulo m.
18. Powers, Roots, and "Unbreakable" Codes.
19. Euler's Phi Function and Sums of Divisors.
20. Powers Modulo p and Primitive Roots.
21. Primitive Roots and Indices.
22. Squares Modulo p.
23. Is -1 a Square Modulo p? Is 2?
24. Quadratic Reciprocity.
25. Which Primes Are Sums of Two Squares?
26. Which Numbers Are Sums of Two Squares?
27. The Equation X^4 + Y^4 = Z^4.
28. Square-Triangular Numbers Revisited.
29. Pell's Equation.
30. Diophantine Approximation.
31. Diophantine Approximation and Pell's Equation.
32. Primality Testing and Carmichael Numbers
33. Number Theory and Imaginary Numbers.
34. The GaussianIntegers and Unique Factorization.
35. Irrational Numbers and Transcendental Numbers.
36. Binomial Coefficients and Pascal's Triangle.
37. Fibonacci's Rabbits and Linear Recurrence Sequences.
38. Generating Functions.
39. Sums of Powers.
40. Cubic Curves and Elliptic Curves.
41. Elliptic Curves with Few Rational Points.
42. Points on Elliptic Curves Modulo p.
43. Torsion Collections Modulo p and Bad Primes.
44. Defect Bounds and Modularity Patterns.
45. Elliptic Curves and Fermat's Last Theorem.
Further Reading.
Appendix A: Factorization of Small Composite Integers.
Appendix B: List of Primes.

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