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A Guide to Elementary Number Theory

Underwood Dudley
Mathematical Association of America
Publication Date: 
Number of Pages: 
Dolciani Mathematical Expositions 41/ MAA Guides 5
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mehdi Hassani
, on

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.

The book contains 39 chapters in 136 pages! So the chapters are short, and they are easy for fast learning and remembering. The titles of chapters are clear and the author the author explains the material very quickly and very clearly, with no extra words. At times there are proofs. Sometimes there are good examples for better understanding. Of course, long proofs (like the proof of Prime Number Theorem) are sketched, and some proofs are dropped.

This book has no exercises, and so it is not a textbook. But I believe that it could actually be used as a concise text book if the instructor adds some details and some exercises. It will be useful for people who need a guide to elementary number theory. High school students and teachers will find some good topics for classroom discussion. Undergraduate students, graduate students and professors will find the book useful for fast reviewing and preparing for exams.

The last chapter of the book reviews some conjectures and unsolved problems in number theory. Among them is the problem of whether odd perfect numbers exist. Professor Dudley’s opinion about this problem, as he has written in his book, is delightful: “…. If one does, it must be very large, and the conjecture is that there is none. My opinion is that there is one — infinitely many, in fact — but it is too large for us to find.”

Mehdi Hassani is a co-tutelle Ph.D. student in Mathematics, Analytic Number Theory in the Institute for Advanced Studies in Basic Science in Zanjan, Iran, and the Université de Bordeaux I, under supervision of the professors M.M. Shahshahani and J-M. Deshouillers.

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. x4 + y4 ≠ z4
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
About the Author