A Guide to Elementary Number Theory Dolciani Mathematical Expositions

__A Guide to Elementary Number Theory__ is a 140 pages exposition of the topics considered in a first course in number theory. It is intended for those who may have seen the material before but have half-forgotten it, and also for those who may have misspent their youth by not having a course in number theory and who want to see what it is about without having to wade through a traditional text, some of which approach 500 pages in length. It will be especially useful to graduate student preparing for the qualifying exams. Though Plato did not quite say, __He is unworthy of the name of man who does not know which integers are the sums of two squares__ he came close. This Guide can make everyone more worthy. A Guide To Elementary Number Theory Is A 140-page Exposition Of The Topics Considered In A First Course In Number Theory. It Is Intended For Those Who May Have Seen The Material Before But Have Half-forgotten It, And Also For Those Who May Have Misspent Their Youth By Not Having A Course In Number Theory And Who Want To See What It Is About Without Having To Wade Through Traditional Texts, Some Of Which Approach 500 Pages In Length. It Will Be Especially Useful To Graduate Students Preparing For Qualifying Exams. Though Plato Did Not Quite Say, He Is Unworthy Of The Name Of Man Who Does Not Know Which Integers Are The Sums Of Two Squares, He Came Close. This Guide Can Make Everyone More Worthy.--page 4 Of Cover. 1. Greatest Common Divisors -- 2. Unique Factorization -- 3. Linear Diophantine Equations -- 4. Congruences -- 5. Linear Congruences -- 6. The Chinese Remainder Theorem -- 7. Fermat's Theorem -- 8. Wilson's Theorem -- 9. The Number Of Divisors Of An Integer -- 10. The Sum Of The Divisors Of An Integer -- 11. Amicable Numbers -- 12. Perfect Numbers -- 13. Euler's Theorem And Function -- 14. Primitive Roots And Orders -- 15. Decimals -- 16. Quadratic Congruences -- 17. Gauss's Lemma -- 18. The Quadratic Reciprocity Theorem -- 19. The Jacobi Symbol -- 20. Pythagorean Triangles -- 21. X4 + Y4 [not Equal] Z4 -- 22. Sums Of Two Squares -- 23. Sums Of Three Squares -- 24. Sums Of Four Squares -- 25. Waring's Problem -- 26. Pell's Equation -- 27. Continued Fractions -- 28. Multigrades -- 29. Carmichael Numbers -- 30. Sophie Germain Primes -- 31. The Group Of Multiplicative Functions -- 32. Bounds For [pi](x) -- 33. The Sum Of The Reciprocals Of The Primes -- 34. The Riemann Hypothesis -- 35. The Prime Number Theorem -- 36. The Abc Conjecture -- 37. Factorization And Testing For Primes -- 38. Algebraic And Transcendental Numbers -- 39. Unsolved Problems. Underwood Dudley. Includes Index. Cover 1 Copyright page 3 Title page 4 Introduction 8 Contents 10 1 Greatest Common Divisors 12 2 Unique Factorization 18 3 Linear Diophantine Equations 22 4 Congruences 24 5 Linear Congruences 28 6 The Chinese Remainder Theorem 32 7 Fermat’s Theorem 36 8 Wilson’s Theorem 38 9 The Number of Divisors of an Integer 40 10 The Sum of the Divisors of an Integer 42 11 Amicable Numbers 44 12 Perfect Numbers 46 13 Euler’s Theorem and Function 48 14 Primitive Roots and Orders 52 15 Decimals 60 16 Quadratic Congruences 62 17 Gauss’s Lemma 68 18 The Quadratic Reciprocity Theorem 72 19 The Jacobi Symbol 78 20 Pythagorean Triangles 82 21 x^4 + y^4 not= z^4 86 22 Sums of Two Squares 90 23 Sums of Three Squares 94 24 Sums of Four Squares 96 25 Waring’s Problem 100 26 Pell’s Equation 102 27 Continued Fractions 106 28 Multigrades 112 29 Carmichael Numbers 114 30 Sophie Germain Primes 116 31 The Group of Multiplicative Functions 118 32 Bounds for pi(x) 122 33 The Sum of the Reciprocals of the Primes 128 34 The Riemann Hypothesis 132 35 The Prime Number Theorem 134 36 The abc Conjecture 136 37 Factorization and Testing for Primes 138 38 Algebraic and Transcendental Numbers 142 39 Unsolved Problems 146 Index 148 About the Author 152 CHOICE Award winner! A Guide to Elementary Number Theory is a 140-page exposition of the topics considered in a first course in number theory. It is intended for those who may have seen the material before but have half-forgotten it, and also for those who may have misspent their youth by not having a course in number theory and who want to see what it is about without having to wade through a traditional text, some of which approach 500 pages in length. It will be especially useful to graduate student preparing for the qualifying exams. Underwood Dudley received the Ph.D. degree (number theory) from the University of Michigan in 1965. He taught at the Ohio State University and at DePauw University, from which he retired in 2004. He is the author of three books on mathematical oddities, The Trisectors, Mathematical Cranks, and Numerology all published by the Mathematical Association of America. He has also served as editor of the College Mathematics Journal, the Pi Mu Epsilon Journal, and two of the Mathematical Association of America's book series A Guide to Elementary Number Theory is a short exposition of the topics considered in a first course in number theory. It is intended for those who have had some exposure to the material before but have half-forgotten it, and also for those who may have never taken a course in number theory but who want to understand it without having to engage with the more traditional texts which are often extensive, and dense. Number theory has an impressive history, which this guide investigates. Rather than being a textbook with exercises and solutions, this guide is an exploration of this interesting and exciting field. Its important results are all included, usually with accompanying the Quadratic Reciprocity Theorem is proved as Gauss did it. The material has been chosen to be maximally broad whilst remaining concise and accessible.