Solved and Unsolved Problems in Number Theory

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. Ttle page PREFACE Chapter I FROM PERFECT NUMBERS TO THE QUADRATIC RECIPROCITY LAW 1. Perfect Numbers 2. Euclid 3. Euler's Converse Proved 4. Euc1id's Algorithm 5. Cataldi and Others 6. The Prime Kumber Theorrm 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 Reciproeity 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 Reriproeity Law a Deep Theorem? 26. Congruential Equations with a Prime Modulus 27. Euler's φ Function 28. Primitive Roots with a Prime Modulus 29. 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 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 √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 5G. A Generalization and Binary Quadratic Forms 57. Some Applications 58 The Significance of Fermat's Equatlon 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 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 STATEMENT ON FUNDAMENTALS TABLE OF DEFINITIONS REFERENCES INDEX From Perfect Numbers To The Quadratic Reciprocity Law -- The Underlying Structure -- Pythagoreanism And Its Many Consequences -- Progress. By Daniel Shanks. Includes Bibliographic References And Index. Many of the basic theorems of number theory -stem from two problems investigated by the Greeksthe problem of perfect numbers and that of Pythagorean numbers.