 Day  Read  Use / Do 
 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
Read some fictional history...
Crack the related "Strap Code"

 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
Get an Encrypted Message

 12 
Fundamental Theorem of Arithmetic

Exercises  The Fundamental Theorem of Arithmetic

 13 
Vigenere's Cipher
Vectors
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 (0224)

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 DiffieHellman Key Exchange Protocol
 
 25 
RSA and Public Key Codes

Exercises  RSA and Public Key Codes

 26 
Finding Phi is Equivalent to Factoring
Finding k^{th} 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)
