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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
|---|---|---|---|---|---|
Searching... | 30000004994145 | QA241 S544 2006 | Open Access Book | Book | Searching... |
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Summary
Summary
Starting with nothing more than basic high school algebra, this volume leads readers gradually from basic algebra to the point of actively performing mathematical research while getting a glimpse of current mathematical frontiers.Features an informal writing style and includes many numerical examples. Emphasizes the methods used for proving theorems rather than specific results. Includes a new chapter on big-Oh notation and how it is used to describe the growth rate of number theoretic functions and to describe the complexity of algorithms. Provides a new chapter that introduces the theory of continued fractions. Includes a new chapter on "Continued Fractions, Square Roots and Pell's Equation." Contains additional historical material, including material on Pell's equation and the Chinese Remainder Theorem.A useful reference for mathematics teachers.
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 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 Euler's 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 Pell's Equation |
| 31 Diophantine Approximation |
| 32 Diophantine Approximation and Pell's Equation |
| 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 Oh, What a Beautiful Function |
| 39 The Topsy-Turvy World of Continued Fractions |
| 40 Continued Fractions, Square Roots and Pell's 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 Fermat's Last Theorem |
| Further Reading |
| A Factorization of Small Composite Integers |
| B A List of Primes |
