Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
Share this page via the icons above, or by copying the link below:
Copy To Clipboard
Successfully Copied!
Solved and Unsolved Problems in Number Theory
 
Daniel Shanks
Solved and Unsolved Problems in Number Theory
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-7645-8
Product Code:  CHEL/297.S
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $62.10
eBook ISBN:  978-1-4704-7650-2
Product Code:  CHEL/297.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $58.50
Softcover ISBN:  978-1-4704-7645-8
eBook: ISBN:  978-1-4704-7650-2
Product Code:  CHEL/297.S.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $120.60 $91.35
Add to cart
Solved and Unsolved Problems in Number Theory
Click above image for expanded view
Solved and Unsolved Problems in Number Theory
Daniel Shanks
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-7645-8
Product Code:  CHEL/297.S
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $62.10
eBook ISBN:  978-1-4704-7650-2
Product Code:  CHEL/297.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $58.50
Softcover ISBN:  978-1-4704-7645-8
eBook ISBN:  978-1-4704-7650-2
Product Code:  CHEL/297.S.B
List Price: $134.00 $101.50
MAA Member Price: $120.60 $91.35
AMS Member Price: $120.60 $91.35
Add to cart
  • Book Details
     
     
    AMS Chelsea Publishing
    Volume: 297; 1985; 305 pp
    MSC: Primary 11

    The investigation of three problems, perfect numbers, periodic decimals, and Pythagorean numbers, has given rise to much of elementary number theory. In this book, Daniel Shanks, past editor of Mathematics of Computation, shows how each result leads to further results and conjectures. The outcome is a most exciting and unusual treatment.

    This edition contains a new chapter presenting research done between 1962 and 1978, emphasizing results that were achieved with the help of computers.
  • Table of Contents
     
     
    • Front Cover
    • CONTENTS
    • PREFACE TO THE THIRD EDITION
    • PREFACE TO THE SECOND EDITION
    • PREFACE TO THE FIRST EDITION
    • CHAPTER I FROM PERFECT NUMBERS TO THE QUADRATIC RECIPROCITY LAW
    • 1. PERFECT NUMBERS
    • 2. EUCLID
    • 3. EULER'S CONVERSE PROVED
    • 4. EUCLID'S ALGORITHM
    • 5. CATALDI AND OTHERS
    • 6. THE PRIME NUMBER THEOREM
    • 7. TWO USEFUL THEOREMS
    • 8. FERMAT AND OTHERS
    • 9. EULER'S GENERALIZATION PROVED
    • 10. PERFECT NUMBERS, II
    • 11. EULER AND M31
    • 12. MANY CONJECTURES AND THEIR INTERRELATIONS
    • 13. SPLITTING THE PRIMES INTO EQUINUMEROUS CLASSES
    • 14. EULER'S CRITERION FORMULATED
    • 15. EULER'S CRITERION PROVED
    • 16. WILSON'S THEOREM
    • 17. GAUSS'S CRITERION
    • 18. THE ORIGINAL LEGENDRE SYMBOL
    • 19. THE RECIPROCITY LAW
    • 20. THE PRIME DIVISORS oF n2 + a
    • CHAPTER II THE UNDERLYING STRUCTURE
    • 21. THE RESIDUE CLASSES AS AN INVENTION
    • 22. THE RESIDUE CLASSES AS A TooL
    • 23. THE RESIDUE CLASSES AS A GROUP
    • 24. QUADRATIC RESIDUES
    • 25. IS THE QUADRATIC RECIPROCITY LAW A DEEP THEOREM?
    • 26. CONGRUENTIAL EQUATIONS WITH A PRIME MODULUS
    • 27. EULER'S Φ FUNCTION
    • 28. PRIMITIVE ROOTs WITH A PRIME MODULUS
    • 29. 𝔐p AS A CYCLIC GROUP
    • 30. THE CIRCULAR PARITY SWITCH
    • 31. PRIMITIVE ROOTS AND FERMAT NUMBERS
    • 32. ARTIN'S CONJECTURES
    • 33. QUESTIONS CONCERNING CYCLE GRAPHS
    • 34. ANSWERS CONCERNING CYCLE GRAPHS
    • 35. FACTOR GENERATORS OF 𝔐m
    • 36. PRIMES IN SOME ARITHMETIC PROGRESSIONS AND A GENERAL DIVISIBILITY THEOREM
    • 37. SCALAR AND VECTOR INDICES
    • 38. THE OTHER RESIDUE CLASSES
    • 39. THE CONVERSE OF FERMAT'S THEOREM
    • 40. SUFFICIENT CONDITIONS FOR PRIMALITY
    • CHAPTER Ill PYTHAGOREANISM AND ITS MANY CONSEQUENCES
    • 41. THE PYTHAGOREANS
    • 42. THE PYTHAGOREAN THEOREM
    • 43. THE √2 AND THE CRISIS
    • 44. THE EFFECT UPON GEOMETRY
    • 45. THE CASE FOR PYTHAGOREANISM
    • 46. THREE GREEK PROBLEMS
    • 47. THREE THEOREMS OF FERMAT
    • 48. FERMAT'S LAST "THEOREM"
    • 49. THE EASY CASE AND INFINITE DESCENT
    • 50. GAUSSIAN INTEGERS AND TWO APPLICATIONS
    • 51. ALGEBRAIC INTEGERS AND KUMMER's THEOREM
    • 52. THE RESTRICTED CASE, SOPHIE GERMAIN, AND WIEFERICH
    • 53. EULER'S "CONJECTURE"
    • 54. SUM OF Two SQUARES
    • 55. A GENERALIZATION AND GEOMETRIC NUMBER THEORY
    • 56. A GENERALIZATION AND BINARY QUADRATIC FORMS
    • 57. SOME APPLICATIONS
    • 58. THE SIGNIFICANCE OF FERMAT'S EQUATION
    • 59. THE MAIN THEOREM
    • 60. AN ALGORITHM
    • 61. CONTINUED FRACTIONS FOR √N
    • 62. FROM ARCHIMEDES TO LUCAS
    • 63. THE LUCAS CRITERION
    • 64. A PROBABILITY ARGUMENT
    • 65. FIBONACCI NUMBERS AND THE ORIGINAL LUCAS TEST
    • SUPPLEMENTARY COMMENTS, THEOREMS, AND EXERCISES
    • CHAPTER IV PROGRESS
    • 66. CHAPTER I FIFTEEN YEARS LATER
    • 67. ARTIN's CONJECTURES, II
    • 68. CYCLE GRAPHS AND RELATED TOPICS
    • 69. PSEUDOPRIMES AND PRIMALITY
    • 70. FERMAT'S LAST "THEOREM," II
    • 71. BINARY QUADRATIC FORMS WITH NEGATIVE DISCRIMINANTS
    • 72. BINARY QUADRATIC FORMS WITH POSITIVE DISCRIMINANTS
    • 73. LUCAS AND PYTHAGORAS
    • 74. THE PROGRESS REPORT CONCLUDED
    • 75. THE SECOND PROGRESS REPORT BEGINS
    • 76. ON JUDGING CONJECTURES
    • 77. ON JUDGING CONJECTURES, II
    • 78. SUB.JECTIVE JUDGEMENT, THE CREATION OF CONJECTURES AND INVENTIONS
    • 79. FERMAT'S LAST "THEOREM," III
    • 80. COMPUTING AND ALGORITHMS
    • 81. 𝒞(3) X 𝒞(3) X 𝒞(3) x 𝒞(3) AND ALL THAT
    • 82. 1993.
    • STATEMENT ON FUNDAMENTALS
    • TABLE OF DEFINITIONS
    • REFERENCES
    • INDEX
    • Back Cover
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
AMS Chelsea Publishing
Volume: 297; 1985; 305 pp
MSC: Primary 11

The investigation of three problems, perfect numbers, periodic decimals, and Pythagorean numbers, has given rise to much of elementary number theory. In this book, Daniel Shanks, past editor of Mathematics of Computation, shows how each result leads to further results and conjectures. The outcome is a most exciting and unusual treatment.

This edition contains a new chapter presenting research done between 1962 and 1978, emphasizing results that were achieved with the help of computers.
  • Front Cover
  • CONTENTS
  • PREFACE TO THE THIRD EDITION
  • PREFACE TO THE SECOND EDITION
  • PREFACE TO THE FIRST EDITION
  • CHAPTER I FROM PERFECT NUMBERS TO THE QUADRATIC RECIPROCITY LAW
  • 1. PERFECT NUMBERS
  • 2. EUCLID
  • 3. EULER'S CONVERSE PROVED
  • 4. EUCLID'S ALGORITHM
  • 5. CATALDI AND OTHERS
  • 6. THE PRIME NUMBER THEOREM
  • 7. TWO USEFUL THEOREMS
  • 8. FERMAT AND OTHERS
  • 9. EULER'S GENERALIZATION PROVED
  • 10. PERFECT NUMBERS, II
  • 11. EULER AND M31
  • 12. MANY CONJECTURES AND THEIR INTERRELATIONS
  • 13. SPLITTING THE PRIMES INTO EQUINUMEROUS CLASSES
  • 14. EULER'S CRITERION FORMULATED
  • 15. EULER'S CRITERION PROVED
  • 16. WILSON'S THEOREM
  • 17. GAUSS'S CRITERION
  • 18. THE ORIGINAL LEGENDRE SYMBOL
  • 19. THE RECIPROCITY LAW
  • 20. THE PRIME DIVISORS oF n2 + a
  • CHAPTER II THE UNDERLYING STRUCTURE
  • 21. THE RESIDUE CLASSES AS AN INVENTION
  • 22. THE RESIDUE CLASSES AS A TooL
  • 23. THE RESIDUE CLASSES AS A GROUP
  • 24. QUADRATIC RESIDUES
  • 25. IS THE QUADRATIC RECIPROCITY LAW A DEEP THEOREM?
  • 26. CONGRUENTIAL EQUATIONS WITH A PRIME MODULUS
  • 27. EULER'S Φ FUNCTION
  • 28. PRIMITIVE ROOTs WITH A PRIME MODULUS
  • 29. 𝔐p AS A CYCLIC GROUP
  • 30. THE CIRCULAR PARITY SWITCH
  • 31. PRIMITIVE ROOTS AND FERMAT NUMBERS
  • 32. ARTIN'S CONJECTURES
  • 33. QUESTIONS CONCERNING CYCLE GRAPHS
  • 34. ANSWERS CONCERNING CYCLE GRAPHS
  • 35. FACTOR GENERATORS OF 𝔐m
  • 36. PRIMES IN SOME ARITHMETIC PROGRESSIONS AND A GENERAL DIVISIBILITY THEOREM
  • 37. SCALAR AND VECTOR INDICES
  • 38. THE OTHER RESIDUE CLASSES
  • 39. THE CONVERSE OF FERMAT'S THEOREM
  • 40. SUFFICIENT CONDITIONS FOR PRIMALITY
  • CHAPTER Ill PYTHAGOREANISM AND ITS MANY CONSEQUENCES
  • 41. THE PYTHAGOREANS
  • 42. THE PYTHAGOREAN THEOREM
  • 43. THE √2 AND THE CRISIS
  • 44. THE EFFECT UPON GEOMETRY
  • 45. THE CASE FOR PYTHAGOREANISM
  • 46. THREE GREEK PROBLEMS
  • 47. THREE THEOREMS OF FERMAT
  • 48. FERMAT'S LAST "THEOREM"
  • 49. THE EASY CASE AND INFINITE DESCENT
  • 50. GAUSSIAN INTEGERS AND TWO APPLICATIONS
  • 51. ALGEBRAIC INTEGERS AND KUMMER's THEOREM
  • 52. THE RESTRICTED CASE, SOPHIE GERMAIN, AND WIEFERICH
  • 53. EULER'S "CONJECTURE"
  • 54. SUM OF Two SQUARES
  • 55. A GENERALIZATION AND GEOMETRIC NUMBER THEORY
  • 56. A GENERALIZATION AND BINARY QUADRATIC FORMS
  • 57. SOME APPLICATIONS
  • 58. THE SIGNIFICANCE OF FERMAT'S EQUATION
  • 59. THE MAIN THEOREM
  • 60. AN ALGORITHM
  • 61. CONTINUED FRACTIONS FOR √N
  • 62. FROM ARCHIMEDES TO LUCAS
  • 63. THE LUCAS CRITERION
  • 64. A PROBABILITY ARGUMENT
  • 65. FIBONACCI NUMBERS AND THE ORIGINAL LUCAS TEST
  • SUPPLEMENTARY COMMENTS, THEOREMS, AND EXERCISES
  • CHAPTER IV PROGRESS
  • 66. CHAPTER I FIFTEEN YEARS LATER
  • 67. ARTIN's CONJECTURES, II
  • 68. CYCLE GRAPHS AND RELATED TOPICS
  • 69. PSEUDOPRIMES AND PRIMALITY
  • 70. FERMAT'S LAST "THEOREM," II
  • 71. BINARY QUADRATIC FORMS WITH NEGATIVE DISCRIMINANTS
  • 72. BINARY QUADRATIC FORMS WITH POSITIVE DISCRIMINANTS
  • 73. LUCAS AND PYTHAGORAS
  • 74. THE PROGRESS REPORT CONCLUDED
  • 75. THE SECOND PROGRESS REPORT BEGINS
  • 76. ON JUDGING CONJECTURES
  • 77. ON JUDGING CONJECTURES, II
  • 78. SUB.JECTIVE JUDGEMENT, THE CREATION OF CONJECTURES AND INVENTIONS
  • 79. FERMAT'S LAST "THEOREM," III
  • 80. COMPUTING AND ALGORITHMS
  • 81. 𝒞(3) X 𝒞(3) X 𝒞(3) x 𝒞(3) AND ALL THAT
  • 82. 1993.
  • STATEMENT ON FUNDAMENTALS
  • TABLE OF DEFINITIONS
  • REFERENCES
  • INDEX
  • Back Cover
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Powered by MathJax
Please select which format for which you are requesting permissions.