A Friendly Introduction to Number Theory (4版)

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. ●50 short chapters provide flexibility and options for instructors and students. A flowchart of chapter dependencies is included in this edition. ●Five basic steps are emphasised throughout the text to help readers develop a robust thought process:。Experimentation 。Pattern recognition 。Hypothesis formation 。Hypothesis testing 。Formal proof ●RSA cryptosystem, elliptic curves, and Fermat's Last Theorem are featured, showing the real-life applications of mathematics.

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 47 The Topsy-Turvy World of Continued Fractions [online] 48 Continued Fractions and Pell's Equation [online] 49 Generating Functions [online] 50 Sums of Powers [online] A Factorization of Small Composite Integers [online] B A List of Primes [online]