Mathematics

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.

• NEW - 8 new chapters—1) Number Theory and Imaginary Numbers; 2) Gaussian Integers and Unique Factorization; 3) Irrational Numbers and Transcendental Numbers; 4) Fibonacci's Rabbits and Linear Recurrence Sequences; 5) Generating Functions; 6) Sums of Powers; 7) Binomial Coefficients and Pascal's Triangle; and 8) Primality Testing and Carmichael Numbers.

• NEW - Many new problems and computer exercises—Over 250 new problems. Problems relocated to the end of each chapter.

  • This provides students with more logical flow of information. Ex.___

• Low key introduction to Number Theory.

  • Allows students to explore an area of math different from standard calculus sequences. Ex.___

• Five basic steps emphasized—Experimentation, pattern recognition, hypothesis formation, hypothesis testing, and formal proof.

  • Encourages students to make mathematical discovers on their own through use of open-ended problems. Ex.___

• RSA cryptosystem, elliptic curves, and Fermat's Last Theorem are featured.

  • Enables students to see real-life applications of mathematics. Ex.___

• Proof of Fermat's Last theorem by Andrew Wiles—Overview provided.

  • Gives students an introduction to one of the most significant mathematical achievements of the 20th century. Ex.___

• 8 new chapters—1) Number Theory and Imaginary Numbers; 2) Gaussian Integers and Unique Factorization; 3) Irrational Numbers and Transcendental Numbers; 4) Fibonacci's Rabbits and Linear Recurrence Sequences; 5) Generating Functions; 6) Sums of Powers; 7) Binomial Coefficients and Pascal's Triangle; and 8) Primality Testing and Carmichael Numbers.

• Many new problems and computer exercises—Over 250 new problems. Problems relocated to the end of each chapter.

  • This provides students with more logical flow of information. Ex.___

 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.

• 
  Friendly Introduction to Number Theory, A, 3/E
  Silverman
  ©2006 | Prentice Hall | Cloth; 448 pp | Instock
  ISBN-10: 0131861379 | ISBN-13: 9780131861374
  Brief Description | Buy from myPearsonStore

  

  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.

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