A Friendly Introduction to Number Theory

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.

Preface

Flowchart of Chapter Dependencies

Introduction

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 and the Chinese Remainder Theorem

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. Primality Testing and Carmichael Numbers

20. Squares Modulo p

21. Quadratic Reciprocity

22. Proof of Quadratic Reciprocity

23. Which Primes Are Sums of Two Squares?

24. Which Numbers Are Sums of Two Squares?

25. Euler’s Phi Function and Sums of Divisors

26. Powers Modulo p and Primitive Roots

27. Primitive Roots and Indices

28. The Equation X4 + Y4 = Z4

29. Square–Triangular Numbers Revisited

30. Pell’s Equation

31. Diophantine Approximation

32. Diophantine Approximation and Pell’s Equation

33. Number Theory and Imaginary Numbers

34. The Gaussian Integers 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. Cubic Curves and Elliptic Curves

39. Elliptic Curves with Few Rational Points

40. Points on Elliptic Curves Modulo p

41. Torsion Collections Modulo p and Bad Primes

42. Defect Bounds and Modularity Patterns

43. Elliptic Curves and Fermat’s Last Theorem

 

Index

 

*44. The Topsy-Turvey World of Continued Fractions [online]

*45. Continued Fractions, Square Roots, and Pell’s Equation [online]

*46. Generating Functions [online]

*47. Sums of Powers [online]

*A. A List of Primes [online]

 

*These chapters are available online.