*A Guide to Elementary Number Theory* is a 140-page exposition of the topics considered in a first course in number theory. It is intended for those who may have seen the material before but have half-forgotten it, and also for those who may have misspent their youth by not having a course in number theory and who want to see what it is about without having to wade through a traditional text, some of which approach 500 pages in length. It will be especially useful to graduate student preparing for the qualifying exams.

Though Plato did not quite say, “He is unworthy of the name of man who does not know which integers are the sums of two squares” he came close. This *Guide* can make everyone more worthy.

### Table of Contents

Introduction

1. Greatest Common Divisors

2. Unique Factorization

3. Linear Diophantine Equations

4. Congruences

5. Linear Congruences

6. The Chinese Remainder Theorem

7. Fermat’s Theorem

8. Wilson’s Theorem

9. The Number of Divisors of an Integer

10. The Sum of the Divisors of an Integer

11. Amicable Numbers

12. Perfect Numbers

13. Euler’s Theorem and Function

14. Primitive Roots and Orders

15. Decimals

16. Quadratic Congruences

17. Gauss's Lemma

18. The Quadratic Reciprocity Theorem

19. The Jacobi Symbol

20. Pythagorean Triangles

21. x^{4} + y^{4} ≠ z^{4}

22. Sums of Two Squares

23. Sums of Three Squares

24. Sums of Four Squares

25. Waring’s Problem

26. Pell’s Equation

27. Continued Fractions

28. Multigrades

29. Carmichael Numbers

30. Sophie Germain Primes

31. The Group of Multiplicative Functions

32. Bounds for *π(x)*

33. The Sum of the Reciprocals of the Primes

34. The Riemann Hypothesis

35. The Prime Number Theorem

36. The *abc* Conjecture

37. Factorization and testing for Primes

38. Algebraic and Transcendental Numbers

39. Unsolved Problems

Index

About the Author

### About the Author

Underwood Dudley received the Ph.D. degree (number theory) from the University of Michigan in 1965. He taught at the Ohio State University and at DePauw University, from which he retired in 2004. He is the author of three books on mathematical oddities, *The Trisectors, Mathematical Cranks*, and *Numerology* all published by the Mathematical Association of America. He has also served as editor of the College Mathematics Journal, the Pi Mu Epsilon Journal, and two of the Mathematical Association of America’s book series.

### MAA Review

Everyone who studies and does mathematics needs, every once in a while, to study or remember some facts of fundamental mathematics, and there is no doubt that we cannot except results and facts of number theory. The main motivation of the author of this book is to provide a friendly volume in response to that need. In fact, the book under review is a concise and useful review of the facts of elementary number theory. It covers most required topics of elementary number theory, and also some strange topics like “Decimals” and “Multigrades,” which are not often found in similar books. Continued...

### All MAA Guides

1. A Guide to Complex Variables

2. A Guide to Advanced Real Analysis

3. A Guide to Real Variables

4. A Guide to Topology

5. A Guide to Elementary Number Theory

6. A Guide to Advanced Linear Algebra

7. A Guide to Plane Algebraic Curves

8. A Guide to Groups, Rings, and Fields

9. A Guide to Functional Analysis