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Publisher:

Mathematical Association of America

Publication Date:

2009

Number of Pages:

139

Format:

Hardcover

Series:

Dolciani Mathematical Expositions 41/ MAA Guides 5

Price:

49.95

ISBN:

9780883853474

Category:

Monograph

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

[Reviewed by , on ]

Mehdi Hassani

03/18/2010

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.

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

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