A Friendly Introduction to Number Theory

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

 

This introductory undergraduate text is designed to entice a wide variety of 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. Emphasis is on the methods used for proving theorems rather than on specific results.

(Pearson)

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1. What Is Number Theory?

2. Pythagorean Triples

3. Pythagorean Triples and the Unit Circle

4. Sums of Higher Powers and Fermats 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 Fermats Little Theorem

10. Congruences, Powers, and Eulers Formula

11. Eulers Phi Function and the Chinese Remainder Theorem

12. Prime Numbers

13. Counting Primes

14. Mersenne Primes

15. Mersenne Primes and Perfect Numbers8

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. Eulers Phi Function and Sums of Divisors

21. Powers Modulo p and Primitive Roots

22. Primitive Roots and Indices

23. Squares Modulo p

24. Is 1 a Square Modulo p? Is 2?

25. Quadratic Reciprocity

26. Which Primes Are Sums of Two Squares?

27. Which Numbers Are Sums of Two Squares?

28. The Equation X4 + Y 4 = Z4

29. Square-Triangular Numbers Revisited

30. Pells Equation

31. Diophantine Approximation

32. Diophantine Approximation and Pells Equation

33. Number Theory and Imaginary Numbers

34. The Gaussian Integers and Unique Factorization

35. Irrational Numbers and Transcendental Numbers

36. Binomial Coefficients and Pascals Triangle

37. Fibonaccis Rabbits and Linear Recurrence Sequences

38. Oh, What a Beautiful Function

39. The Topsy-Turvy World of Continued Fractions

40. Continued Fractions, Square Roots and Pells Equation

41. Generating Functions

42. Sums of Powers

43. Cubic Curves and Elliptic Curves

44. Elliptic Curves with Few Rational Points

45. Points on Elliptic Curves Modulo p

46. Torsion Collections Modulo p and Bad Primes

47. Defect Bounds and Modularity Patterns

48. Elliptic Curves and Fermats Last Theorem

Further Reading

A. Factorization of Small Composite Integers

B. A List of Primes

(Pearson)