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Three Pearls of Number Theory (Dover Books on Mathematics) by A. Y. Khinchin
These 3 puzzles involve the proof of a basic law governing the world of numbers known to be correct in all tested cases — the problem is to prove that the law is always correct. Includes van der Waerden's theorem on arithmetic progressions, the Landau-Schnirelmann hypothesis and Mann's theorem, and a solution to Waring's problem.
- GenresMathematics
Unknown Binding
First published February 18, 1998
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About the author
Aleksandr Yakovlevich Khinchin
10 books2 followersАлекса́ндр Я́ковлевич Хи́нчин
A.I. Khinchin
A.Y. Khinchin
A. Ya. Khinchin
A.I. Khinchin
A.Y. Khinchin
A. Ya. Khinchin
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Displaying 1 - 4 of 4 reviews
February 5, 2017
Khinchin wrote this book as a letter to a former student of his, wounded during WWII and willing to spend out his recovery with some deep theorems in Arithmetic (Number Theory) whose proofs were “elementary” —not necessarily “simple”.
The aforementioned theorems are van der Waerden’s result on arithmetic progressions, the Landau-Schnirelmann hypothesis and Artin and Scherk’s proof of Mann’s theorem concerning densities of sequences of positive integers, and last, but not least, Waring’s problem.
The exposition goes in order of increasing complexity, so the first “chapter” on van der Waerden’s theorem is accessible to the largest audience (college algebra will suffice), the next will be more challenging (background in exact sciences or engineering is strongly recommended to get the best out of it), but the last chapter, an elementary solution to Waring’s problem, is by far the most demanding of them all —a useful piece of advice is to work by yourself certain, if not all, calculations in this chapter when it comes to the proof of the “fundamental lemma”—.
All in all, Khinchin stands out as a great expositor in this tiny little book, and does a lot of justice to each of these three pearls in Number Theory. An invaluable book.
The aforementioned theorems are van der Waerden’s result on arithmetic progressions, the Landau-Schnirelmann hypothesis and Artin and Scherk’s proof of Mann’s theorem concerning densities of sequences of positive integers, and last, but not least, Waring’s problem.
The exposition goes in order of increasing complexity, so the first “chapter” on van der Waerden’s theorem is accessible to the largest audience (college algebra will suffice), the next will be more challenging (background in exact sciences or engineering is strongly recommended to get the best out of it), but the last chapter, an elementary solution to Waring’s problem, is by far the most demanding of them all —a useful piece of advice is to work by yourself certain, if not all, calculations in this chapter when it comes to the proof of the “fundamental lemma”—.
All in all, Khinchin stands out as a great expositor in this tiny little book, and does a lot of justice to each of these three pearls in Number Theory. An invaluable book.
September 30, 2017
According to the opening preface of this book, Khinchin sent these Three Pearls of Number Theory to a former student recuperating during World War II. This is all the backstory that we are given for this. Khinchin claims that any schoolboy should be able to understand this stuff, but does admit that it is quite deep. It’s not that I don’t try to understand, but when I see the long lines of text that are supposed to denote numbers my attention wanders. It is rather shameful for me to say this. I guess a lack of structure really does have a bad part to it.
Anyway, this book was quite short; it measures a meager sixty-four pages in length. I would like to be able to understand what is going on with the proofs and all of that, so maybe I should revisit Velleman’s How To Prove It.
Anyway, this book was quite short; it measures a meager sixty-four pages in length. I would like to be able to understand what is going on with the proofs and all of that, so maybe I should revisit Velleman’s How To Prove It.
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May 18, 2010Description of three theorems from number theory.
January 1, 2023
Three theorems from combinatorial number theory, proved using nothing more sophisticated than induction. In lieu of an intro Khinchin has written a letter to a (fictional? real?) convalescent Soviet soldier, healing at the end of WW2. This soldier, apparently an *extremely* diligent former student of the author, has written him to ask for some math to think of while he recovers from his heroic injuries. Khinchin salutes the courage of his correspondent's generation, which he says compares quite favorably with that of the WW1 generation. This might be genuine, might be something to appease the censors (who famously could not hack their way through most of what the Soviet mathematicians produced). But it's a very charming conceit, to imagine ruddy-cheeked Sergei dutifully checking all the hypotheses and cases for these theorems.
The first is van der Waerden's theorem that it you partition a sufficiently long segment of the positive integers into a fixed number of classes, one of the classes is guaranteed to contain an arithmetic progression. The proof here is I think fairly standard, cropping up in combinatorics and discrete math classes as well. Already the reader will have to create some of their own diagrams to keep track of all the moving pieces. "Keeping track" might be the motto for this entire book, because with all the theorems and mental apparatus of math basically unavailable the reader has to do a lot of basically logistical work: remembering what things are, comparing quantities, etc.
The second result, Mann's theorem on the superadditivity of lower asymptotic density, has probably the prettiest proof. All the preliminary basics from Landau and Schnirelmann would make I think very enjoyable reading for a math amateur as well. The construction here is so delicate yet so natural. Khinchin also provides a lot of good historical background (true also for Chapter 1), having been involved in the quest to prove the theorem himself.
Chapter 3 is the hardest result by far: the elementary solution by Linnik of Waring's problem. This was to show that, no matter what n is, if we add together k copies (k depending on n) of the sequence of n-th powers of natural numbers: 0^n, 1^n, 2^n, 3^n, ... then we obtain all of the natural numbers. Here we see something similar to Chapter 2: reduction of the crux to a technical lemma. Khinchin in both cases has his dessert first, drawing the theorem from the lemma. The real hard work is reserved for proving the technical lemma, and here it is very hard work indeed. Nothing is, conceptually speaking, very deep, but a great deal of reduction has to be gone through. Another road bump in here is the free-floating usage of constants that depend on a parameter (c(n)), which can be multiplied together and added without changing in notation. This is probably familiar to people who have studied big-O notation in computer science but might take some getting used to for complete novices. (I would nit-pick the way that Khinchin presents one of his arrays at one point, switching his indices around slightly.) By the time you come to the end of this proof, even if you're a seasoned mathematician, you might be pretty tired and have a hard time remembering what the point of it all was. Thankfully Khinchin includes a reassuring valediction for his wounded soldier friend: "This proof, so exquisitely elementary, will undoubtedly seem very complicated to you. But it will take you only two to three weeks' work with pencil and paper to understand and digest it completely. It is by conquering difficulties of just this sort, that the mathematician grows and develops."
I think that if someone wanted to seriously study combinatorial number theory (which hadn't been so named when this little pamphlet was published) then a more recent book would serve better. But as a historical artifact it's hard to ignore the charms of Khinchin's little book, which packs so much into such a tiny frame. If you really want to test your mastery of basic discrete math, this is a fine read.
The first is van der Waerden's theorem that it you partition a sufficiently long segment of the positive integers into a fixed number of classes, one of the classes is guaranteed to contain an arithmetic progression. The proof here is I think fairly standard, cropping up in combinatorics and discrete math classes as well. Already the reader will have to create some of their own diagrams to keep track of all the moving pieces. "Keeping track" might be the motto for this entire book, because with all the theorems and mental apparatus of math basically unavailable the reader has to do a lot of basically logistical work: remembering what things are, comparing quantities, etc.
The second result, Mann's theorem on the superadditivity of lower asymptotic density, has probably the prettiest proof. All the preliminary basics from Landau and Schnirelmann would make I think very enjoyable reading for a math amateur as well. The construction here is so delicate yet so natural. Khinchin also provides a lot of good historical background (true also for Chapter 1), having been involved in the quest to prove the theorem himself.
Chapter 3 is the hardest result by far: the elementary solution by Linnik of Waring's problem. This was to show that, no matter what n is, if we add together k copies (k depending on n) of the sequence of n-th powers of natural numbers: 0^n, 1^n, 2^n, 3^n, ... then we obtain all of the natural numbers. Here we see something similar to Chapter 2: reduction of the crux to a technical lemma. Khinchin in both cases has his dessert first, drawing the theorem from the lemma. The real hard work is reserved for proving the technical lemma, and here it is very hard work indeed. Nothing is, conceptually speaking, very deep, but a great deal of reduction has to be gone through. Another road bump in here is the free-floating usage of constants that depend on a parameter (c(n)), which can be multiplied together and added without changing in notation. This is probably familiar to people who have studied big-O notation in computer science but might take some getting used to for complete novices. (I would nit-pick the way that Khinchin presents one of his arrays at one point, switching his indices around slightly.) By the time you come to the end of this proof, even if you're a seasoned mathematician, you might be pretty tired and have a hard time remembering what the point of it all was. Thankfully Khinchin includes a reassuring valediction for his wounded soldier friend: "This proof, so exquisitely elementary, will undoubtedly seem very complicated to you. But it will take you only two to three weeks' work with pencil and paper to understand and digest it completely. It is by conquering difficulties of just this sort, that the mathematician grows and develops."
I think that if someone wanted to seriously study combinatorial number theory (which hadn't been so named when this little pamphlet was published) then a more recent book would serve better. But as a historical artifact it's hard to ignore the charms of Khinchin's little book, which packs so much into such a tiny frame. If you really want to test your mastery of basic discrete math, this is a fine read.
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