Number theory: a historical approach

Machine generated contents note: 1.Number Theory Begins Pierre de Fermat Pythagorean Triangles Babylonian Mathematics Sexagesimal Numbers Regular Numbers Square Numbers Primitive Pythagorean Triples Infinite Descent Arithmetic Progressions Fibonacci's Approach Problems 2.Euclid Greek Mathematics Triangular Numbers Tetrahedral and Pyramidal Numbers The Axiomatic Method Proof by Contradiction Euclid's Self-Evident Truths Unique Factorization Pythagorean Tuning Problems 3.Divisibility The Euclidean Algorithm The Greatest Common Divisor The Division Algorithm Divisibility The Fundamental Theorem of Arithmetic Congruences Divisibility Tests Continued Fractions Problems 4.Diophantus The Arithmetica Problems from the Arithmetica A Note in the Margin Diophantine Equations Pell's Equation Continued Fractions Problems 5.Fermat Christmas Day, 1640

Contents note continued: Fermat's Little Theorem Primes as Sums of Two Squares Sums of Two Squares Perfect Numbers Mersenne Primes Fermat Numbers Binomial Coefficients "Multi Pertransibunt et Augebitur Scientia" Problems 6.Congruences Fermat's Little Theorem Linear Congruences Inverses The Chinese Remainder Theorem Wilson's Theorem Two Quadratic Congruences Lagrange's Theorem Problems 7.Euler and Lagrange A New Beginning Euler's Phi Function Primitive Roots Euler's Identity Quadratic Residues Lagrange Lagrange's Four Squares Theorem Sums of Three Squares Waring's Problem Fermat's Last Theorem Problems 8.Gauss The Young Gauss Quadratic Residues The Legendre Symbol Euler's Criterion Gauss's Lemma Euler's Conjecture The Law of Quadratic Reciprocity Problems 9.Primes I Factoring The Quadratic Sieve Method Is n Prime? Pseudoprimes Absolute Pseudoprimes

Contents note continued: A Probabilistic Test Can n Divide 2n1 or 2n+1? Mersenne Primes Problems 10.Primes II Gaps Both Large and Small The Twin Prime Conjecture The Series Bertrand's Postulate Goldbach's Conjecture Arithmetic Progressions Problems 11.Sophie Germain Monsieur LeBlanc Germain Primes Germain's Grand Plan Fermat's Last Theorem Problems 12.Fibonacci Numbers Fibonacci The Fibonacci Sequence The Golden Ratio Fibonacci Numbers in Nature Binet's Formula Tiling and the Fibonacci Numbers Fibonacci Numbers and Divisibility Generating Functions Problems 13.Cryptography Secret Codes on Mount Everest Caesar and Vigenere Ciphers Unbreakable Ciphers Public-Key Systems Problems 14.Continued Fractions The Golden Ratio Revisited Finite Continued Fractions Infinite Continued Fractions Approximation Pell's Equation Problems 15.Partitions Euler

Contents note continued: Generating Functions

Euler's Pentagonal Number Theorem

Ferrers Graphs

Ramanujan

Problems

Hints for Selected Problems

Solutions to Selected Problems

Brief Introduction to Sage

Suggestions for Further Reading

Pronunciation Guide

Table of Primes.