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A Friendly Introduction to Number Theory

Joseph H. Silverman
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Miklós Bóna
, on

The title is certainly accurate. The author treats the main areas of number theory in a leisurely pace. For instance, prime numbers are first discussed in Chapter 12. The irrationality of the square root of two comes around in Chapter 37. This comfortable speed implies that there are many ways in which one can teach a course from the book, since not every chapter depends on the preceding one. There are helpful diagrams in the introduction explaining the possibilities.

There are significant changes from the previous edition. On the one hand, the Quadratic Reciprocity Theorem is now fully proved, which means the addition of a new Chapter 23, which is much more difficult than the chapters leading to it. There is also a chapter on induction, which will be welcome by student who did not have a transition to higher math course before. On the other hand, and this is rare, the author wanted to keep the book to a managable size, and therefore he removed chapters 47–50 from the print edition. These are freely available online.

The one thing I missed in this book is full solutions to some of the exercises and hints for some others. Students taking this class will typically be rather new to theorem proving; they would have benefited from some sample solutions. If you think that your students will not mind their absence, then the book is probably a good choice for you.

Miklós Bóna is Professor of Mathematics at the University of Florida.


Flowchart of Chapter Dependencies


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. Is -1 a Square Modulo p? Is 2?

22. Quadratic Reciprocity

23. Proof of Quadratic Reciprocity

24. Which Primes Are Sums of Two Squares?

25. Which Numbers Are Sums of Two Squares?

26. As Easy as One, Two, Three

27. Euler’s Phi Function and Sums of Divisors

28. Powers Modulo p and Primitive Roots

29. Primitive Roots and Indices

30. The Equation X4 + Y4 = Z4

31. Square–Triangular Numbers Revisited

32. Pell’s Equation

33. Diophantine Approximation

34. Diophantine Approximation and Pell’s Equation

35. Number Theory and Imaginary Numbers

36. The Gaussian Integers and Unique Factorization

37. Irrational Numbers and Transcendental Numbers

38. Binomial Coefficients and Pascal’s Triangle

39. Fibonacci’s Rabbits and Linear Recurrence Sequences

40. Oh, What a Beautiful Function

41. Cubic Curves and Elliptic Curves

42. Elliptic Curves with Few Rational Points

43. Points on Elliptic Curves Modulo p

44. Torsion Collections Modulo p and Bad Primes

45. Defect Bounds and Modularity Patterns

46. Elliptic Curves and Fermat’s Last Theorem

Further Reading


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

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

*49. Generating Functions [online]

*50. Sums of Powers [online]

*A. Factorization of Small Composite Integers [online]

*B. A List of Primes [online]

*These chapters are available online.