AMS Chelsea Publishing 1985; 305 pp; hardcover Volume: 297 Reprint/Revision History: reprinted 1993; first AMS printing 2001 ISBN10: 082182824X ISBN13: 9780821828243 List Price: US$44 Member Price: US$39.60 Order Code: CHEL/297.H
 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 IIII  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
