Often the saying is true, and you cannot judge a book by its title, let alone its cover. However, this is not the case with the book under review, as De Koninck and Mercier’s book is exactly what its name suggests: a collection of 1001 problems that cover the breadth of classical number theory, with sections of problems dedicated to induction, divisibility, prime numbers, congruences, Diophantine equations, quadratic reciprocity, and continued fractions, among others. The book does open with a dozen pages dedicated to setting up notation and giving definitions and introducing the reader to some of the ideas that will be used in the problems, but quickly gets to the heart of the matter: the 1001 problems, starting with
1. Show that for each positive integer n we have ∑1≤k≤n k = ½n(n+1)
(where we have changed the notation slightly to accomodate the limitations of html) and ending with
1001. Does there exist a rational number x so that
π 5x + 2π4x2+ 3π3x3+ 4π2x4+ 5πx5+ 6 = 0?
After the problems come the solutions to all 1001 problems, which are written up in a clear, easy to follow style and often contain bibliographic or historical information about a problem.
The problems range in difficulty from problems that any alumnus of a class in elementary number theory should be able to do in their sleep, through problems from various math competitions and the kinds of problems one would find in Mathematics Magazine, to problems that professional number theorists will struggle to figure out. One interesting — or frustrating, depending on your point of view — choice that the authors made was to give no indication of the difficulty or the source of their problem when stating the problem, and only telling you where the problem came from once you have decided to give up and look at the solution. The problems generally increase in difficulty as you go through the book, but this is far from uniform, and some of the easier problems come late in a chapter.
There are other choices the authors make that will appeal to some readers and discourage other readers — many of the problems require the use of computers to solve, and the authors certainly let their fondness for arithmetical functions shine through while other topics one might hope for are ignored — but the book is littered with gems, and one can imagine it being a great addition to sit around any math lounge or library where it can be picked up and flipped through. I also know that my copy will have a place near my desk the next time I teach a number theory course to serve as inspiration!
Darren Glass (email@example.com) is an Assistant Professor at Gettysburg College.