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Three Pearls of Number Theory (Dover Books on Mathematics)

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It is truly a pearl, and pearls have permanence; -- they retain their beauty, from one generation to the next.

So too is the case for this little book. Measured in mathematical generations, you must count back a few;-- back to the last year of the Second World War, and in what was then The Soviet Union; now Russia. The author, A. Y. Khinchin was (and is) a mathematical physicist of World Renown. He has seminal contributions to number theory, to statistics, to information theory, and to statistical physics.

The book is unique in many ways; for one, I believe it is for everyone, -- even if you don't know math. But readers with math background will know that it is possible for writing in math to be both moving and beautiful. This is the case for this little classic. Both the historical background and the subject are unique.

The nature of the book (64 pages in all!) is almost like a personal letter written by a loving teacher to one of his students, but it is much more than that.

At the time, the War had devastated Russia, and almost everyone from the young generation, including students of the sciences was at the front. The casualties everywhere in The Soviet Union were staggering; many had lost parents and relatives during 4 long years of destruction.

Khinchin's student Seryozha was recovering (at the time of the letter) in an army hospital, and he had written his former teacher, asking for math problems to work on. We can't begin to imagine the terrible conditions of army hospitals on the front at this time. The care Khinchin took in responding is moving. In fact Seryozha had only taken one or two beginning classes at university, before being sent to war. And even though Khinchin had only a vague recollection of Seryozha from a class, he truly wanted to send him something he could use, -- something that would make him happy. Students at the front were giving their lives for the rest of the country, and we must remember that this was a war where the difference between good and evil was crystal clear. Khinchin's students were heroes. The book opens with a moving and personal letter, full of empathy, gratitude and love.

As for the mathematics, Khinchin had carefully selected problems of great beauty, problems that can be stated and appreciated with little specialized knowledge; -- in modern lingo, with very few prerequisites. And at the same time, they are problems Seryozha can work on in his hospital bed. They are profound, and they can be attacked with elementary means. Naturally, since 1945, there have been a lot of advances on all three. The problems are from arithmetic (or number theory), and they go under the names: (a) van der Waerden's theorem on arithmetic progressions, (b) Landau's hypothesis and Mann's theorem, and (c) an elementary solution of Waring's problem.

By now these three problems take a different form in modern math books, but none as beautiful, in my opinion as Khinchin's in his loving letter to his student written toward the end of the war. Review by Palle Jorgensen, May 2005.

This book is actually a letter from a Russian professor to a student sent off to war. It's short, but won't be an easy read. These are "pearls" but getting the oyster open is going to be tough. It's also remarkable for it's candid revelation of the mathematical process of professional practitioners at various universities in different countries. The first pearl is about a young student name van der Waerden. Yep, the guy who went on to prove so many results in Abstract Algebra and wrote the classic text on the subject influencing Artin and Noether. It's interesting to note, van der Waerden used finite differences in his proof recounted in the first pearl, and he's the only author I know that included finite differences in his abstract algebra text book. Both the candid historical confessions and the conversational exposition make this a great book. It's style and methods should be widely imitated. Come on, professors, write more like this! Future archaeologist of the 20th century will be glad this document is available for it's revelation of the habits of homo professorus mathematicus.

A Y Khinchin was one of the greatest mathematicians of the first half of the twentieth century. He was also famous as a teacher and communicator. Fortunately, several of the books he wrote are still in print in English translations, published by Dover. Like William Feller and Richard Feynman he combines a complete mastery of his subject with an ability to explain clearly without sacrificing mathematical rigour.

This is a short book of three chapters: Chapter 1. Van der Waerden's theorem on arithmetic progressions. Chapter 2. The Landau-Shnirelmann hypothesis and Mann's theorem. Chapter 3. An elementary solution of Waring's problem.

These are all difficult problems from the theory of numbers and I think that the elementary proofs that Khinchin describes here are original. This book is a challenging but enjoyable read.

I also recommend his other book on number theory: "Continued Fractions".

A jewel. Highly recommended for students of number theory, expecilally amateurs and teachers.