Problems and Proofs in Numbers and Algebra

This is a text for a bridge or transition-to-proofs course, but aimed at pre-service middle-school teachers and bright high-school students rather than sophomore math majors. The mathematical level is therefore much lower than we are used to in this kind of text. How low? Well, even though it is a proofs course, the proof of the Fundamental Theorem of Arithmetic (unique factorization of integers into primes) is judged to be too advanced (p. 7), so it is taken as an axiom.

The discussion of proof techniques is very informal, and there’s no mathematical logic. There’s also no mathematical induction. The exercises are reasonable, although almost none of them are to prove things; they concentrate on applying theorems in the text to specific numerical problems.

The book is divided into two halves. The Numbers half is elementary number theory, with a little bit of combinatorics. The Algebra half is the basis for high-school algebra: roots and factorization of polynomials, and some material on systems of linear equations.

Overall the treatment is clear and simple, although there are some rough spots. The RSA cryptosystem is given as an example, and the algorithm is described well, but there’s no indication of why it works mathematically (Euler’s extension of Fermat’s little theorem) or why it is secure (difficulty of factoring large numbers). There’s a nice but obsolete discussion of the Universal Product Code (UPC) and the International Standard Book Number (ISBN). These are described as 12 and 10 digit codes respectively, although a quick glance at the book’s back cover shows a 13-digit bar code and a 13-digit ISBN, sure to confuse the curious reader. (The book’s discussion is for the older UPC-A and ISBN-10 encodings, not the more current EAN and ISBN-13.)

I think the most serious omission, especially for students at this level, is that there is little discussion of experimentation in mathematics. For example, on p. 9 we are asked to find all integers \(x\) such that \(x^4 + 4\) is prime. This is a very good problem, but the solution omits the obvious step of trying the first few values \(x=1, 2, 3, \dots\) to see which ones give primes. Instead the book decides, based on no evidence, that \(x^4 + 4\) factors into two quadratics and goes through a non-obvious method to find these. If you instead work out several examples, it’s not too hard to guess that the expression does factor, and only a little harder to guess what the factorization is.

Very Good Feature: two extended Euclidean algorithms for finding the greatest common divisor as a linear combination of the terms, that works as you go, rather than back-substituting all the way up at the end. (The two extended algorithms are actually the same, but formalized differently; they’re called here the tableau method and the matrix method.)

Very Bad Feature: no index.

Bottom line: a good but very limited book, useful to its target audience.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.