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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.
Author Bio
Silverman, Joseph H : Brown University
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 X^4 + Y^4 = Z^4
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
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.
Author Bio
Silverman, Joseph H : Brown University
Table of Contents
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 X^4 + Y^4 = Z^4
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