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Preface | p. v |
Flowchart of Chapter Dependencies | p. ix |
Introduction | p. 1 |
What Is Number Theory? | p. 6 |
Pythagorean Triples | p. 13 |
Pythagorean Triples and the Unit Circle | p. 21 |
Sums of Higher Powers and Fermat's Last Theorem | p. 26 |
Divisibility and the Greatest Common Divisor | p. 30 |
Linear Equations and the Greatest Common Divisor | p. 37 |
Factorization and the Fundamental Theorem of Arithmetic | p. 46 |
Congruences | p. 55 |
Congruences, Powers, and Fermat's Little Theorem | p. 65 |
Congruences, Powers, and Euler's Formula | p. 71 |
Euler's Phi Function and the Chinese Remainder Theorem | p. 75 |
Prime Numbers | p. 83 |
Counting Primes | p. 90 |
Mersenne Primes | p. 96 |
Mersenne Primes and Perfect Numbers | p. 101 |
Powers Modulo m and Successive Squaring | p. 111 |
Computing k^{th} Roots Modulo m | p. 118 |
Powers, Roots, and "Unbreakable" Codes | p. 123 |
Primality Testing and Carmichael Numbers | p. 129 |
Squares Modulo p | p. 141 |
Is-1 a Square Modulo p? Is 2? | p. 148 |
Quadratic Reciprocity | p. 159 |
Proof of Quadratic Reciprocity | p. 171 |
Which Primes Are Sums of Two Squares? | p. 181 |
Which Numbers Are Sums of Two Squares? | p. 193 |
As Easy as One, Two, Three | p. 199 |
Euler's Phi Function and Sums of Divisors | p. 206 |
Powers Modulo p and Primitive Roots | p. 211 |
Primitive Roots and Indices | p. 224 |
The Equation X^{4}+Y^{4}=Z^{4} | p. 231 |
Square-Triangular Numbers Revisited | p. 236 |
Pell's Equation | p. 245 |
Diophantine Approximation | p. 251 |
Diophantine Approximation and Pell's Equation | p. 260 |
Number Theory and Imaginary Numbers | p. 267 |
The Gaussian Integers and Unique Factorization | p. 281 |
Irrational Numbers and Transcendental Numbers | p. 297 |
Binomial Coefficients and Pascal's Triangle | p. 313 |
Fibonacci's Rabbits and Linear Recurrence Sequences | p. 324 |
Oh, What a Beautiful Function | p. 339 |
Cubic Curves and Elliptic Curves | p. 353 |
Elliptic Curves with Few Rational Points | p. 366 |
Points on Elliptic Curves Modulo p | p. 373 |
Torsion Collections Modulo p and Bad Primes | p. 384 |
Defect Bounds and Modularity Patterns | p. 388 |
Elliptic Curves and Fermat's Last Theorem | p. 394 |
Further Reading | p. 396 |
Index | p. 397 |
The Topsy-Turvy World of Continued Fractions [online] | p. 410 |
Continued Fractions and Pell's Equation [online] | p. 426 |
Generating Functions [online] | p. 442 |
Sums of Powers [online] | p. 452 |
Factorization of Small Composite Integers [online] | p. 464 |
A List of Primes [online] | p. 466 |
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