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Applied Number Theory

Harald Niederreiter and Arne Winterhof
Publication Date: 
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Underwood Dudley
, on

The authors’ first chapter is a quick review of number theory. There follow chapters on four applications: cryptography, coding theory, quasi-Monte Carlo methods, and pseudorandom numbers. A final chapter touches on other applications, for example how quantum computers, when they arrive, can solve the hidden subgroup problem.

The book is intended to be usable as a text — there are exercises at the ends of chapters, but with no solutions — and the authors mention “undergraduates,” though it would be a rare undergraduate curriculum that contains a suitable course and a rare undergraduate who would master the material. Graduate courses in applied number theory are probably thin on the ground, so I hope the book will cause some to be created. It would also be useful for those of us who received our training in number theory in the days before it had any applications and wanted to find out about them. And, of course, for anyone else, of whatever age or experience, who was curious.

The writing is clear and lively. There is a list of more than two hundred references. The book is pleasant to the eyes, and in the hands. I noticed no typographical errors, a tribute to Springer’s maintaining standards in these days when it seems as if some publishers have laid off all their copy editors. It is an estimable book that I hope is successful.

Now to the bad part. The authors start chapters with what they call limericks, but aren’t. They have the aabba rhyme scheme, but violate the rules on scansion and stress. E.g.,

This theory of the old Greeks,
the first mathematical geeks,
is a marvel of charm and beauty,
it’s number theory, this cutie,
and we are its devoted freaks.

However, authors who can quote Tom Stoppard, who wrote in Rosencrantz and Guildenstern Are Dead that “The sun came up about as often as it went down, in the long run” can be forgiven a lot and should be allowed latitude.

Woody Dudley’s first number theory teacher did what some would call a horrible job, in effect copying the textbook onto the blackboard. But the material was so gorgeous that he went on, and on.

See the table of contents in the publisher's webpage.