This little book — it reaches 64 pages only if we count the title page and the Library of Congress information — is a Dover reprint of a 1952 translation of the 1946 edition of a book published in Russian in 1945. It contains statements and proofs of three topics in elementary number theory: Van der Waerden’s theorem on arithmetic progressions, the Landau-Schnirelman hypothesis, and Waring’s problem. These are three great topics, but in the twenty-first century this book is no longer the best resource for any of these.
The proof of the Landau-Schnirelman hypothesis (Mann's theorem) has been redone even more simply than it is here. (See for example Narkiewicz's Number Theory.) There's been a lot of work on van der Waerden's theorem over the last sixty years, and again there are now simpler proofs. (See for example the survey of combinatorial number theory by Pomerance and Sárközy in Handbook of Combinatorics.) Linnik’s solution to Waring's problem can be found in a similar (but easier-to-understand) form in Nathanson's Elementary Methods in Number Theory.
For all of these reasons, together with the archaic-looking typeface and lack of an index, this reviewer was prepared to dismiss this book out of hand.
And yet, and yet.
It turns out that Three Pearls comes with a back story. The author, Khinchin, received a letter from a soldier near the Russian front lines in the second world war. The soldier, identified only as “Seryozha” had been laid up in the hospital, and to pass the time wanted something mathematical to study. He thought of his former professor, whose lectures he had attended for just a year, and wrote to Khinchin with this request. (Khinchin was clearly a teacher who left an impression on his students!) Given this request, Khinchin chose three problems in elementary number theory, all of which had elementary (though difficult) solutions, and all of which had been recently solved by somebody who was not a “great number theorist”.
The book is written in the form of a lengthy epistle, and Khinchin gently guides Seryozha (and by extension, us) through the proofs with helpful examples, clarifications, and helpful notes on notation. The style is beautiful and friendly — I found myself reading more closely than I had intended to. This book would not be easily readable by most undergraduates, but I can easily imagine a strong student reading this with considerable pleasure, and having her interest in number theory sparked forever.
Dominic Klyve is an Assistant Professor of Mathematics at Central Washington University and a founder and director of the Euler Archive. He divides his (work) time between studying elementary number theory, studying the history of mathematics, and wondering whether the student to whom Khinchin addressed his book worked through all three proofs.