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Edition: 2ND 01

Copyright: 2001

Publisher: Prentice Hall, Inc.

Published: 2001

International: No

Copyright: 2001

Publisher: Prentice Hall, Inc.

Published: 2001

International: No

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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.

**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

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Summary

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

Publisher Info

Publisher: Prentice Hall, Inc.

Published: 2001

International: No

Published: 2001

International: No

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.

**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