Manfred Schroeder received his Ph.D. in physics from the University of Göttingen in 1954; his thesis described resonance properties of concert halls. He spent much of his career at Bell Labs in New Jersey. Schroeder expects a reader of Number Theory in Science and Communication to have a background like his own, or like that of his younger colleagues at Bell Labs: the reader should be someone who is mathematically curious and enjoys writing programs to look for patterns, but does not have a strong background in pure mathematics. In particular, Schroeder assumes that his readers are familiar with the Fourier transform, and can gain intuition for the behavior of functions by examining their Fourier transforms, but he does not require any prior experience with abstract algebra. Perhaps the ideal reader is an electrical engineering student who enjoyed her math courses. Schroeder discusses a long list of applications and recreations related to number theory, including Mersenne primes, types of cryptography, error-correcting codes, acoustics, and fractals. The tone is informal, conversational, and enthusiastic.
Schroeder’s treatment can be superficial or scattered. The reader who needs proofs should not expect to find them in this book, but even the reader who prefers illustrative computations may want supplementary reading. For example, Schroeder praises continued-fraction expansions of irrational numbers, but doesn’t explain how to manipulate them, or how to obtain the expansions that he uses. Informal discussions of fields are spread throughout the book. Section 1.5 describes some finite fields and gives the multiplication table for a field of order 4, but the definition of a field and the fact that the integers modulo 4 are not a field do not appear until Section 6.2. There is no entry for field in the index.
Number Theory in Science and Communication was first published in 1984; the edition under review is the fifth, dated 2009. Number theory has been a very active field in the last twenty-seven years, and Schroeder’s text has a palimpsest quality, with later mathematical advances layered on earlier ones. In the section on Fermat’s Last Theorem, for instance, one paragraph begins, “If the proof of FLT has proved so difficult, perhaps the theorem is just not true,” the next begins, “In the meantime, the ‘impossible’ has happened and Fermat’s Last Theorem has been proved,” and the following paragraph reads “In the summer of 1997 the Göttingen Academy of Sciences, after due consideration of the published proof, is expected to award Wiles the Wolfskehl Prize, which now amounts to about 70,000 Marks.” (The prize was indeed awarded!)
Number Theory in Science and Communication is rewarding to browse, or as a jumping-off point for further research, but does not suffice as a stand-alone text. It would be a good source of student projects in an undergraduate discrete mathematics or number theory course. Students could pick a topic (quantum cryptography? Coin tossing by telephone?), do some research to discover the current state of the art, and use a computer to work out illustrative examples.
Ursula Whitcher is a Teaching and Research Postdoctoral Fellow at Harvey Mudd College.
Introduction.- The Natural Numbers.- Primes.- The Prime Distribution.- Fractions: Continued, Egyptian and Farey.- Linear Congruences.- Diophantine Equations.- The Theorems of Fermat, Wilson and Euler.- Permutation, Cycles and Derangements.- Euler Trap Doors and Public-Key Encryption.- The Divisor Functions.- The Prime Divisor Functions.- Certified Signatures.- Primitive Roots.- Knapsack Encryption.- Quadratic Residues.- The Chinese Remainder Theorem and Simultaneous Congruences.- Fast Transformations and Kronecker Products.- Quadratic Congruences.- Psudoprimes, Poker and Remote Coin Tossing.- The Möbius Function and the Möbius Transform.- From Error Correction Codes to Covering Sets.- Generating Functions and Partitions.- Cyclotomic Polynomials.- Linear Systems and Polynomials.- Polynomial Theory.- Galois Fields.- Spectral Properties of Galois Sequences.- Random Number Generators.- Waveforms and Radiation Patterns.- Number Theory, Randomness and "Art".