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# Three Pearls of Number Theory

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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.  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.

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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.  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.

## 38 review for Three Pearls of Number Theory

1. 4 out of 5

Francisco Paniagua

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. 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.

2. 5 out of 5

William Schram

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 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.

3. 5 out of 5

Boaz

Description of three theorems from number theory.

4. 5 out of 5

Gwyn Whieldon

5. 4 out of 5

Fred Katz

6. 5 out of 5

James Nickel

7. 4 out of 5

Emiliano Kargieman

8. 5 out of 5

Stephen

9. 4 out of 5

Adarsh Mishra

10. 4 out of 5

Kaiser

11. 4 out of 5

Richard

12. 5 out of 5

saleh

13. 4 out of 5

Hasn

14. 5 out of 5

Trincadour

15. 4 out of 5

Jovany Agathe

16. 4 out of 5

Fabius Bonnet

17. 4 out of 5

MathMonk

18. 4 out of 5

Kennedy Corrêa

19. 4 out of 5

Rza Nuriyev

20. 5 out of 5

Jordan

21. 4 out of 5

Liam

22. 5 out of 5

Brian Ball

23. 5 out of 5

Steve

24. 4 out of 5

Franco

25. 5 out of 5

Jeff Kowalski

26. 5 out of 5

Biju

27. 5 out of 5

Homoionym

28. 5 out of 5

Charles

29. 4 out of 5

Marco Spadini

30. 4 out of 5

Diueine Monteiro

31. 4 out of 5

Rod Oliveira

32. 4 out of 5

Min

33. 4 out of 5

Evan

34. 5 out of 5

juanja

35. 5 out of 5

Jones

36. 5 out of 5

Aakansh Gupta

37. 5 out of 5

Sandra

38. 4 out of 5

Rogelio Yoyontzin

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