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
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
*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.