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Solved and Unsolved Problems in Number Theory
Daniel Shanks
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AMS Chelsea Publishing
1985; 305 pp; hardcover
Volume: 297
Reprint/Revision History:
reprinted 1993; first AMS printing 2001
ISBN-10: 0-8218-2824-X
ISBN-13: 978-0-8218-2824-3
List Price: US$41
Member Price: US$36.90
Order Code: CHEL/297.H
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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

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 \(M_{31}\)
  • 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 \(n^2 +a\)
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 \(\phi\) function
  • 28 Primitive roots with a prime modulus
  • 29 \(\mathfrak{M}_{p}\) 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 \(\mathfrak{M}_{m}\)
  • 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 III: Pythagoreanism and Its Many Consequences
  • 41 The Pythagoreans
  • 42 The Pythagorean theorem
  • 43 The \(\sqrt 2\) 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 \(\sqrt N\)
  • 62 From Archimedes to Lucas
  • 63 The Lucas criterion
  • 64 A probability argument
  • 65 Fibonacci numbers and the original Lucas test
Appendix to Chapters I-III
  • 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 Subjective judgement, the creation of conjectures and inventions
  • 79 Fermat's last "Theorem," III
  • 80 Computing and algorithms
  • 81 \(\scr{C}(3)\times\scr{C}(3)\times\scr{C}(3)\times\scr{C}(3)\) and all that
  • 82 1993
Appendix
  • Statement on fundamentals
  • Table of definitions
  • References
  • Index
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