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. Number Theory Title Page 1 Contents 2 Preface to the second edition 4 Preface to the first edition 5 From perfect numbers... 7 Perfect Numbers 7 Euclid 9 Euler's Converse Method 11 Euclid's Algorithm 11 Cataldi and others 13 The Prime Number Theorem 14 Two useful theorems 15 Fermat and Others 16 Euler's Generalization Proved 18 Perfect Numbers II 19 Euler and M31 19 Many Conjectures and Their Interrelations 21 Splitting the Primes into Equinumerous Classes 22 Euler's Criterion Formulated 23 Euler's Criterion Proved 24 Wilson's Theorem 25 Gauss's Criterion 26 The Original Legendre Symbol 27 The Reciprocity Law 28 The Prime Divisors of n^2+a 30 Appendix to Chapters I-III 107 Underlying structure 32 The Residue Classes as an Invention 32 Pythagoreanism... 67 Supplementary exercices 107 Progress 115 Statements on fundamentals 126 Table of definitions 128 References 130 Index 142 Number,Pythagoreanism,Number Theory,Conjectures 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.