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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.
Provides a low key introduction to Number Theory.
Encourages students to make mathematical discoveries on their own through five basic steps: experimentation, pattern recognition, hypothesis formation, hypothesis testing, and formal proof.
Covers material not usually presented at this level, such as the RSA cryptosystem, elliptic curves, and an overview of Wiles' proof of Fermat's Last Theorem.
All the core text problems are built into the text and must be worked by all students.
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. Cubic Curves and Elliptic Curves.
33. Elliptic Curves with Few Rational Points.
34. Points on Elliptic Curves Modulo p.
35. Torsion Collections Modulo p and Bad Primes.
36. Defect Bounds and Modularity Patterns.
37. Elliptic Curves and Fermat's Last Theorem.
Appendix A: Factorization of Small Composite Integers.
Appendix B: List of Primes.
Additional Exercises.
Index.
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.
Provides a low key introduction to Number Theory.
Encourages students to make mathematical discoveries on their own through five basic steps: experimentation, pattern recognition, hypothesis formation, hypothesis testing, and formal proof.
Covers material not usually presented at this level, such as the RSA cryptosystem, elliptic curves, and an overview of Wiles' proof of Fermat's Last Theorem.
All the core text problems are built into the text and must be worked by all students.
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. Cubic Curves and Elliptic Curves.
33. Elliptic Curves with Few Rational Points.
34. Points on Elliptic Curves Modulo p.
35. Torsion Collections Modulo p and Bad Primes.
36. Defect Bounds and Modularity Patterns.
37. Elliptic Curves and Fermat's Last Theorem.
Appendix A: Factorization of Small Composite Integers.
Appendix B: List of Primes.
Additional Exercises.
Index.