A Friendly Introduction to Number Theory Joseph H Silverman

A Friendly Introduction to Number Theory par Joseph H Silverman

A Friendly Introduction to Number Theory Joseph H Silverman

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Résumé

For courses in Elementary Number Theory for non-maths majors, Mathematics/Number Theory for mathematics education students, Number Theory and Computer Science. This is an introductory undergraduate text designed to entice non-maths majors into learning some mathematics, while teaching them to think mathematically at the same time.

A Friendly Introduction to Number Theory Joseph H Silverman

For courses in Elementary Number Theory for non-math majors, for mathematics education students, and for Computer Science students.

This is an introductory undergraduate text designed to entice non-math majors into learning some mathematics, while teaching them to think mathematically at the same time. Starting with nothing more than basic high school algebra, the reader is gradually led from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers. The writing style is informal and includes many numerical examples, which are analyzed for patterns and used to make conjectures. The emphasis is on the methods used for proving theorems rather than on specific results.

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.

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. Euler's Phi Function and Sums of Divisors.

20. Powers Modulo p and Primitive Roots.

21. Primitive Roots and Indices.

22. Squares Modulo p.

23. Is -1 a Square Modulo p? Is 2?

24. Quadratic Reciprocity.

25. Which Primes Are Sums of Two Squares?

26. Which Numbers Are Sums of Two Squares?

27. The Equation X4 + Y4 = Z4.

28. Square-Triangular Numbers Revisited.

29. Pell's Equation.

30. Diophantine Approximation.

31. Diophantine Approximation and Pell's Equation.

32. Primality Testing and Carmichael Numbers

33. Number Theory and Imaginary Numbers.

34. The Gaussian Integers and Unique Factorization.

35. Irrational Numbers and Transcendental Numbers.

36. Binomial Coefficients and Pascal's Triangle.

37. Fibonacci's Rabbits and Linear Recurrence Sequences.

38. Generating Functions.

39. Sums of Powers.

40. Cubic Curves and Elliptic Curves.

41. Elliptic Curves with Few Rational Points.

42. Points on Elliptic Curves Modulo p.

43. Torsion Collections Modulo p and Bad Primes.

44. Defect Bounds and Modularity Patterns.

45. Elliptic Curves and Fermat's Last Theorem.

Further Reading.

Appendix A: Factorization of Small Composite Integers.

Appendix B: List of Primes.

Index.

A Friendly Introduction to Number Theory Joseph H Silverman

A Friendly Introduction to Number Theory Joseph H Silverman

Résumé

A Friendly Introduction to Number Theory Résumé

A Friendly Introduction to Number Theory Joseph H Silverman

Sommaire

Informations supplémentaires