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Some Preliminary Considerations | |
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Mathematical Induction | |
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The Binomial Theorem | |
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Early Number Theory | |
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Divisibility Theory in the Integers | |
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The Division Algorithm | |
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The Greatest Common Divisor | |
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The Euclidean Algorithm | |
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The Diophantine Equation ax+by=c | |
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Primes and Their Distribution | |
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The Fundamental Theorem of Arithmetic | |
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The Sieve of Eratosthenes | |
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The Goldbach Conjecture | |
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The Theory of Congruences | |
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Carl Friedrich Gauss | |
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Basic Properties of Congruence | |
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Special Divisibility Tests | |
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Linear Congruences | |
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Fermat's Theorem | |
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Pierre de Fermat | |
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Fermat's Factorization Method | |
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The Little Theorem | |
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Wilson's Theorem | |
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Number-Theoretic Functions | |
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The Functions ��nd �ã | |
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The Mobius Inversion Formula | |
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The Greatest Integer Function | |
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An Application to the Calendar | |
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Euler's Generalization of Fermat's Theorem | |
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Leonhard Euler | |
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Euler's Phi-Function | |
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Euler's Theorem | |
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Some Properties of the Phi-Function | |
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An Application to Cryptography | |
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Primitive Roots and Indices | |
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The Order of an Integer Modulo n | |
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Primitive Roots for Primes | |
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Composite Numbers Having Prime Roots | |
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The Theory of Indices | |
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The Quadratic Reciprocity Law | |
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Euler's Criterion | |
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The Legendre Symbol and Its Properties | |
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Quadratic Reciprocity | |
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Quadratic Congruences with Composite Moduli | |
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Perfect Numbers | |
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The Search for Perfect Numbers | |
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Mersenne Primes | |
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Fermat Numbers | |
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The Fermat Conjecture | |
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Pythagorean Triples | |
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The Famous ��Last Theorem�� | |
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Representation of Integers as Sums of Squares | |
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Joseph Louis Lagrange | |
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Sums of Two Squares | |
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Sums of More than Two Squares | |
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Fibonacci Numbers | |
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The Fibonacci Sequence | |
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Certain Identities Involving Fibonacci Numbers | |
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Continued Fractions | |
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Srinivasa Ramanujan | |
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Finite Continued Fractions | |
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Infinite Continued Fractions | |
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Pell's Equation | |
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Some Twentieth-Century Developments | |
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Hardy, Dickson, and Erdos | |
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Primality Testing and Factorization | |
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An Application to Factoring: Remote Coin-Flipping | |
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The Prime Number Theorem | |