This book is a very idiosyncratic introductory text in number theory. The author’s starting point is the statement, “Much of elementary number theory arose out of the investigation of three problems; that of perfect numbers, that of periodic decimals, and that of Pythagorean numbers. We have accordingly organized the book into three long chapters.” (p. xi) The book expands and weaves together the ideas arising in these three areas to give a fairly comprehensive coverage of elementary number theory. Each problem leads to more problems, some solved and some still unsolved. It is not a problem book, but a book that uses problems to drive the exposition.

This book is now in its fourth edition (the first was in 1962, the fourth in 1993), and another idiosyncratic feature is that it was updated not by revising the text but by adding a series of supplements; these now make up one-third of the book. The supplements deal with the same problems as in the main work, but usually in more depth, and they cover what were (at time of publication) some of the latest results. My favorite part is the long discussion on the nature of conjectures beginning on p. 239. Shanks thought there should be very strong evidence in favor of a statement before it is dignified by the name “conjecture”. This book is also the source of his famous statement (regarding experimental evidence for the non-existence of odd perfect numbers), “10^{50} is a long way from infinity”. (p. 217)

Overall I think this approach didn’t work well for an introductory text; the idea is good, but the execution jumps around too much and will bewilder the beginning student. That said, for the professional mathematician it is fascinating to see the interconnections in these many topics. A good introductory text where consideration of particular problems and numerical evidence drive the exposition is R. P. Burn’s A Pathway Into Number Theory.

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at MathNerds.org, a math help site that fosters inquiry learning.