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1 Closure and Sets of Numbers
Remainders and Divisibility
2 Direct Arguments
Proof by Cases
Exercises - Divisibility
3 Indirect Argument (Proof by Contradiction)

Secret Decoder Rings and the Caeser Cipher
Simple Transposition Ciphers
(English) Letter Frequencies
Encryptor / Decryptor
Exercises - Indirect Arguments
Exercises - The Pigeonhole Principle

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4 Summations and Sigma Notation
Two Important Properties of Sums (Linearity)
The Principle of Mathematical Induction
Exercises - Induction and Sums
5 Variations on Induction
Proof of the Division Algorithm
Exercises - Induction in Other Contexts
6 Patterns and Conjectures
Generating Pascal's Triangle
An Odd Question About Pascal's Triangle
Fibonacci's Rabbits
Numbers in Different Bases
A Curious Inequality
Exercises - Patterns and Conjectures
Exercises - The Collatz Conjecture
7 Pythagorean Triples Exercises - Pythagorean Triples
8 Permutation Puzzles (and Transposition Ciphers Revisited)
Jug Problems
Exercises - Permutation Puzzles
9 The Euclidean Algorithm
Lame's Theorem
Exercises - The Euclidean Algorithm
Gcd Finder
10 Linear Combinations Exercises - Linear Combinations
Linear Combination Finder
11 Congruence
Solving Linear Congruences
Affine Ciphers
Exercises - Linear Congruences

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12 Fundamental Theorem of Arithmetic Exercises - The Fundamental Theorem of Arithmetic
13 Vigenere's Cipher
Dot Products, Norms, and Angles Between Vectors
Exercises - Vigenere's Cipher
Vigenere Subsequence Analyzer
Review Exercises - Review Exercises (Set A)
14 Linear Transformations
Inverses of Linear Transformations
Exercises - Linear Transformations
15 The Hill Cipher Exercises - The Hill Cipher
16 Continued Fractions Exercises - Continued Fractions
17 Wilson's Theorem
Converse of Wilson's Theorem
Exercises - Wilson's Theorem
18 Fast Exponentiation
Fermat's Little Theorem
Exercises - Fast Exponentiation and Fermat's Little Theorem
19 Primality Testing and Carmichael Numbers
Euler's Theorem
20 Euler's Phi Function
Table of Phi Values (0-224)
Exercises - Euler's Theorem
21 Euler's Phi and the Chinese Remainder Theorem
Solving a System of Congruences
Exercises - Chinese Remainder Theorem
22 Cardinality and Infinite Sets
Enumeration of the (Positive) Rationals
Cantor's Diagonal Argument
Exercises - Cardinality and Infinite Sets
23 The Cardinality of a Power Set
Review Exercises - Review Exercises (Set B)
24 The Diffie-Hellman Key Exchange Protocol
25 RSA and Public Key Codes
Exercises - RSA and Public Key Codes
26 Finding Phi is Equivalent to Factoring
Finding kth Roots (mod n)
27 Finding Square Roots (mod n) Exercises - Finding Roots
28 Finding Certain Roots is Equivalent to Factoring
29 Flipping Coins over the Telephone
30 The Order of an Integer
Primitive Roots
Index Arithmetic
Exercises - The Order of an Integer and Primitive Roots
Exercises - Index Arithmetic
31 Power Residues and Legendre's Symbol
Properties of Legendre's Symbol
Exercises - Legendre's Symbol
32 Gauss' Lemma
A Consequence of Gauss' Lemma
33 Quadratic Reciprocity
Review Exercises - Review Exercises (Set C)