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A Friendly Introduction to Number Theory

Joseph H. Silverman
Pearson Prentice Hall
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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In the August/September 2003 issue of FOCUS, I wrote an article entitled "What's the Best Textbook — Elementary Number Theory," which I hoped would be the first of many such articles by many authors. (It didn't work.) In that article, I chose this book as the best textbook for (my version of) the Elementary Number Theory course, in part because this is "a book that works hard to be approachable." That is important for me, because the course I teach at Colby is intended to be accessible to a wide range of students. And, as I also explained, readability is important: I want my students to be able to read their textbook.

Of course, readability is not enough. The content must be right too. In an elementary book, I usually look for hints that the author understands the deeper parts of the subject. Most of my students are not yet able to decide by themselves what is deep and what is routine, what is a brilliant insight and what is just technical prowess. An author who really understands his subject can put up signposts to help students grasp such things. By doing so, he can also keep the book interesting for those students in my class who already have quite a bit of mathematical background.

On all of these measures, Silverman's book is excellent. It is insightful, interesting, and fun. From my article:

Silverman does a great job of exactly those things I care about: his book is readable; it makes students aware of the role of both experimentation and proof; it includes many pointers to deeper questions and even to open questions. Even the jokes are the sort that I would make!

What's more, my students seem to agree. They do not find the book easy, but they do seem to be able to read it and get something out of their reading.

Of course, no book is perfect. For me, this book's biggest fault is the fact that it doesn’t include a proof of quadratic reciprocity. This, of course, is easy to fix: I supply one on a handout, usually based on the one given in Dan Flath's Introduction to Number Theory.

Any author who calls his mathematics textbook "friendly" is being quite daring. Silverman pulls it off.

As far as I can tell, this new edition does not include all that many significant changes. Good old problem 21.3, which has always seemed very sneaky to me, is still there. So are all the neat chapters at the end, which I never seem to have time for. Maybe this year.

Fernando Q. Gouvêa teaches at Colby College in Waterville, ME.

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 Numbers8

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. Euler’s Phi Function and Sums of Divisors

21. Powers Modulo p and Primitive Roots

22. Primitive Roots and Indices

23. Squares Modulo p

24. Is —1 a Square Modulo p? Is 2?

25. Quadratic Reciprocity

26. Which Primes Are Sums of Two Squares?

27. Which Numbers Are Sums of Two Squares?

28. The Equation X4 + Y 4 = 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. Oh, What a Beautiful Function

39. The Topsy-Turvy World of Continued Fractions

40. Continued Fractions, Square Roots and Pell’s Equation

41. Generating Functions

42. Sums of Powers

43. Cubic Curves and Elliptic Curves

44. Elliptic Curves with Few Rational Points

45. Points on Elliptic Curves Modulo p

46. Torsion Collections Modulo p and Bad Primes

47. Defect Bounds and Modularity Patterns

48. Elliptic Curves and Fermat’s Last Theorem

Further Reading

A. Factorization of Small Composite Integers

B. A List of Primes