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Full Description
A Friendly Introduction to Number Theory, Fourth Edition is designed to introduce readers 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, readers 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 analyzed for patterns and used to make conjectures. Emphasis is on the methods used for proving theorems rather than on specific results.
Contents
PrefaceFlowchart of Chapter DependenciesIntroduction1. What Is Number Theory?2. Pythagorean Triples3. Pythagorean Triples and the Unit Circle4. Sums of Higher Powers and Fermat's Last Theorem5. Divisibility and the Greatest Common Divisor6. Linear Equations and the Greatest Common Divisor7. Factorization and the Fundamental Theorem of Arithmetic8. Congruences9. Congruences, Powers, and Fermat's Little Theorem10. Congruences, Powers, and Euler's Formula11. Euler's Phi Function and the Chinese Remainder Theorem12. Prime Numbers13. Counting Primes14. Mersenne Primes15. Mersenne Primes and Perfect Numbers16. Powers Modulo m and Successive Squaring17. Computing kth Roots Modulo m18. Powers, Roots, and "Unbreakable" Codes19. Primality Testing and Carmichael Numbers20. Squares Modulo p21. Quadratic Reciprocity22. Proof of Quadratic Reciprocity23. Which Primes Are Sums of Two Squares?24. Which Numbers Are Sums of Two Squares?25. Euler's Phi Function and Sums of Divisors26. Powers Modulo p and Primitive Roots27. Primitive Roots and Indices28. The Equation X4 + Y4 = Z429. Square-Triangular Numbers Revisited30. Pell's Equation31. Diophantine Approximation32. Diophantine Approximation and Pell's Equation33. Number Theory and Imaginary Numbers34. The Gaussian Integers and Unique Factorization35. Irrational Numbers and Transcendental Numbers36. Binomial Coefficients and Pascal's Triangle37. Fibonacci's Rabbits and Linear Recurrence Sequences38. Cubic Curves and Elliptic Curves39. Elliptic Curves with Few Rational Points40. Points on Elliptic Curves Modulo p41. Torsion Collections Modulo p and Bad Primes42. Defect Bounds and Modularity Patterns43. Elliptic Curves and Fermat's Last TheoremIndex*44. The Topsy-Turvey World of Continued Fractions [online]*45. Continued Fractions, Square Roots, and Pell's Equation [online]*46. Generating Functions [online]*47. Sums of Powers [online]*A. A List of Primes [online]*These chapters are available online.
