Friendly Introduction to Number Theory, A (Classic Version): (Pearson Modern Classics for Advanced Mathematics Series)

  For one-semester undergraduate courses in Elementary Number Theory     A Friendly Introduction to Number Theory, 4th Edition is designed to introduce students to the overall themes and methodology of mathematics through the detailed study of one particular facet–number theory. Starting with nothing more than basic high school algebra, students are gradually led to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing is appropriate for the undergraduate audience and includes many numerical examples, which are analysed for patterns and used to make conjectures. Emphasis is on the methods used for proving theorems rather than on specific results.

Table of Contents:
Preface Flowchart of Chapter Dependencies Introduction 1. What Is Number Theory? 2. Pythagorean Triples 3. Pythagorean Triples and the Unit Circle 4. Sums of Higher Powers and Fermat’s Last Theorem 5. Divisibility and the Greatest Common Divisor 6. Linear Equations and the Greatest Common Divisor 7. Factorization and the Fundamental Theorem of Arithmetic 8. Congruences 9. Congruences, Powers, and Fermat’s Little Theorem 10. Congruences, Powers, and Euler’s Formula 11. Euler’s Phi Function and the Chinese Remainder Theorem 12. Prime Numbers 13. Counting Primes 14. Mersenne Primes 15. Mersenne Primes and Perfect Numbers 16. Powers Modulo m and Successive Squaring 17. Computing kth Roots Modulo m 18. Powers, Roots, and “Unbreakable” Codes 19. Primality Testing and Carmichael Numbers 20. Squares Modulo p 21. Is -1 a Square Modulo p? Is 2? 22. Quadratic Reciprocity 23. Proof of Quadratic Reciprocity 24. Which Primes Are Sums of Two Squares? 25. Which Numbers Are Sums of Two Squares? 26. As Easy as One, Two, Three 27. Euler’s Phi Function and Sums of Divisors 28. Powers Modulo p and Primitive Roots 29. Primitive Roots and Indices 30. The Equation X4 + Y4 = Z4 31. Square–Triangular Numbers Revisited 32. Pell’s Equation 33. Diophantine Approximation 34. Diophantine Approximation and Pell’s Equation 35. Number Theory and Imaginary Numbers 36. The Gaussian Integers and Unique Factorization 37. Irrational Numbers and Transcendental Numbers 38. Binomial Coefficients and Pascal’s Triangle 39. Fibonacci’s Rabbits and Linear Recurrence Sequences 40. Oh, What a Beautiful Function 41. Cubic Curves and Elliptic Curves 42. Elliptic Curves with Few Rational Points 43. Points on Elliptic Curves Modulo p 44. Torsion Collections Modulo p and Bad Primes 45. Defect Bounds and Modularity Patterns 46. Elliptic Curves and Fermat’s Last Theorem    

About the Author :
Joseph H. Silverman is a Professor of Mathematics at Brown University. He received his Sc.B. at Brown and his Ph.D. at Harvard, after which he held positions at MIT and Boston University before joining the Brown faculty in 1988. He has published more than 100 peer-reviewed research articles and seven books in the fields of number theory, elliptic curves, arithmetic geometry, arithmetic dynamical systems, and cryptography.  He is a highly regarded teacher, having won teaching awards from Brown University and the Mathematical Association of America, as well as a Steele Prize for Mathematical Exposition from the American Mathematical Society. He has supervised the theses of more than 25 Ph.D. students, is a co-founder of NTRU Cryptosystems, Inc., and has served as an elected member of the American Mathematical Society Council and Executive Committee.