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| Preface | |
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| Flowchart of Chapter Dependencies | |
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| Introduction | |
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| What Is Number Theory? | |
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| Pythagorean Triples | |
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| Pythagorean Triples and the Unit Circle | |
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| Sums of Higher Powers and Fermat's Last Theorem | |
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| Divisibility and the Greatest Common Divisor | |
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| Linear Equations and the Greatest Common Divisor | |
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| Factorization and the Fundamental Theorem of Arithmetic | |
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| Congruences | |
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| Congruences, Powers, and Fermat's Little Theorem | |
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| Congruences, Powers, and Euler's Formula | |
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| Euler's Phi Function and the Chinese Remainder Theorem | |
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| Prime Numbers | |
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| Counting Primes | |
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| Mersenne Primes | |
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| Mersenne Primes and Perfect Numbers | |
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| Powers Modulo m and Successive Squaring | |
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| Computing k<sup>th</sup> Roots Modulo m | |
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| Powers, Roots, and "Unbreakable" Codes | |
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| Primality Testing and Carmichael Numbers | |
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| Squares Modulo p | |
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| Is-1 a Square Modulo p? Is 2? | |
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| Quadratic Reciprocity | |
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| Proof of Quadratic Reciprocity | |
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| Which Primes Are Sums of Two Squares? | |
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| Which Numbers Are Sums of Two Squares? | |
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| As Easy as One, Two, Three | |
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| Euler's Phi Function and Sums of Divisors | |
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| Powers Modulo p and Primitive Roots | |
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| Primitive Roots and Indices | |
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| The Equation X<sup>4</sup>+Y<sup>4</sup>=Z<sup>4</sup> | |
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| Square-Triangular Numbers Revisited | |
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| Pell's Equation | |
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| Diophantine Approximation | |
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| Diophantine Approximation and Pell's Equation | |
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| Number Theory and Imaginary Numbers | |
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| The Gaussian Integers and Unique Factorization | |
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| Irrational Numbers and Transcendental Numbers | |
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| Binomial Coefficients and Pascal's Triangle | |
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| Fibonacci's Rabbits and Linear Recurrence Sequences | |
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| Oh, What a Beautiful Function | |
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| Cubic Curves and Elliptic Curves | |
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| Elliptic Curves with Few Rational Points | |
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| Points on Elliptic Curves Modulo p | |
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| Torsion Collections Modulo p and Bad Primes | |
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| Defect Bounds and Modularity Patterns | |
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| Elliptic Curves and Fermat's Last Theorem | |
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| Further Reading | |
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| Index | |
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| The Topsy-Turvy World of Continued Fractions [online] | |
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| Continued Fractions and Pell's Equation [online] | |
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| Generating Functions [online] | |
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| Sums of Powers [online] | |
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| Factorization of Small Composite Integers [online] | |
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| A List of Primes [online] | |