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Number Theory

Róbert Freud and Edit Gyarmati
Publication Date: 
Number of Pages: 
Pure and Applied Undergraduate Texts
[Reviewed by
Karl-Dieter Crisman
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Number Theory by Freud and Gyarmati is the latest entry in the signature "Sally series" of undergraduate textbooks from the AMS.  The website blurb says, "The book is suitable for both graduate and undergraduate courses with enough material to fill two or more semesters and could be used as a source for independent study and capstone projects."  In a nutshell, that is pretty accurate; indeed, this is an English version (not just translation) of a textbook used in several editions throughout Hungary.  
But what sort of courses?
The list of topics in the first half of the book is perhaps surprisingly conventional, but this belies both the presentation and exercises.  A hint that it will not be a typical mass-market US number theory text is given already in the first chapter, where the concept of "special common divisor" (essentially, gcd up to units) is given prominence - in preparation for cases where (even in ordered rings) ordering/"greatest" will not necessarily be a useful concept.  Inclusion of not just the Miller-Rabin test, but the Solovay-Strassen primality test, or that the sum of primes up to n is asymptotic to n^2/(2 log(n)), are similar indications.  Some of my favorite parts were an intuitive rationale for the formula for the Waring number, and a very entertaining snippet of a lecture by Erdös, a friend of both authors.
However, if the reader of this review were to start thinking the text is simply extremely terse, or simply has a very high expectation of prerequisite knowledge, that is not accurate either.  This is not comparable in sophistication to (say) Apostol's Analytic Number Theory text, and certainly not to Hardy.  I would instead say that this text can be brief, and would succeed best with students with high aptitude for assimilation of concepts, but for such students the examples are appropriate, and (importantly) they do not need particularly deep background at all.  Calculus connections when they occur, for example, would be easy to avoid or use as black boxes. Although later in the text there is a characterization of the algebraic integers of quadratic number fields, as well as proving the Fundamental Theorem of Arithmetic for unique factorization domains, these topics are reasonably situated and (by that point) they should be well prepared for them.
Nonetheless, elementary doesn't mean easy, and for the "typical" course at a "typical" institution in North America, this would be a challenging text to use.  However, if one were to replace a standard two-semester algebra course (if your institution has that) with a course developed from this text, that might be a good - or even better - replacement.  The clever naming system for parts of examples and proofs are helpful (though numbering of examples would be helpful too); the many places where multiple proofs of a result are given, usually with indication of why one might want multiple proofs, are useful; and the substantial answers and hints given to exercises soften some but not all of the blow of their difficulty.  
For a one-term course with ambitious students, Chapters 1 and 2 are, as the authors say, foundational; using the first few sections each in Chapters 3-7 and then picking a few bonus ones as desired (including from Chapters 8, 9, and 12) could work well. For an independent study on combinatorial number theory or algebraic number theory without tears, the later sections are also recommended.  As always with such texts, for an instructor using some other text, asking your library to order it so you can mine it for interesting connections is a fine idea.
Karl-Dieter Crisman is Professor of Mathematics at Gordon College.  When asked to come up with a course for majors as a new faculty member, he immediately jumped at the chance to teach number theory, which led to an abiding interest in using open source software to explore this deep subject with students in class.