This is a deceptive book. If you just look at the narrative, it appears to be a simple, maybe too simple, introduction to elementary number theory. But really the narrative just prepares you to work on the exercises, which although elementary are very challenging. The book includes complete solutions to all the exercises, so you can use it either as a problem book or as a topics book. This book was a Dover semi-original, being published by them in 1954 as an English translation of the fifth Russian edition from 1949, and reprinted several times, most recently in 2016.

The exercises extend the ideas in the body, often by a long distance. For example, the floor (greatest integer) function is discussed in the body, and is used for all kinds of counting problems in the exercises. There are also quite a number of problems involving trigonometric sums (one of the author’s research specialties), which are presented as extensions of congruence facts and the Chinese Remainder Theorem. The basics of Pell’s equation are done as exercises, starting from the theory of quadratic residues. The most startling inclusion is the Pólya-Vinogradov inequality in the special case of the Legendre symbol.

For the most part the exercises are broken down in small chunks, so even difficult results should be solvable by diligent students. A few look like they might be very difficult even then. Unlike the body, many of the exercises deal with estimates and inequalities, and there is some asymptotic reasoning.

Very Bad Feature: no index. There are also no citations and no bibliographic material.

Another good book with very good exercises is Niven & Zuckerman & Montgomery’s An Introduction to the Theory of Numbers. This is a considerably more advanced and comprehensive book that Vinogradov’s, and it only gives hints and not complete solutions.

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