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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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