Number Theory (with an emphasis on Cryptology)

Notes

  1. Closure and Sets of Numbers
  2. Remainders and Divisibility
  3. Induction & The Well-Ordering Principle
  4. Proof of the Division Algorithm
  5. Summations and Sigma Notation
  6. Two Important Properties of Sums
  7. Fibonacci's Rabbits
  8. Numbers in Different Bases
  9. Pythagorean Triples
  10. The GCD and the Euclidean Algorithm
  11. Exploring the Euclidean Algorithm with Mathematica
  12. Congruences
  13. The Fundamental Theorem of Arithmetic
  14. Continued Fractions
  15. Wilson's Theorem
  16. The Converse of Wilson's Theorem
  17. Fast Exponentiation
  18. Fermat's Little Theorem
  19. Euler's Theorem
  20. Euler's Phi Function and the Chinese Remainder Theorem
  21. Solving a System of Congruences Simultaneously
  22. Finding kth Roots (Mod n)
  23. The Order of an Integer
  24. Primitive Roots
  25. Index Arithmetic
  26. Power Residues and Legendre's Symbol
  27. Properties of Legendre's Symbol

Connections

  1. Vectors, Linear Transformations, and Matrix Forms
  2. Inverses of Linear Transformations
  3. Dot Products, Norms, and Angles Between Vectors
  4. Cardinality and Infinite Sets
  5. The Enumeration of the Positive Rationals
  6. Cantor's Diagonal Argument
  7. The Cardinality of the Power Set
  8. The Taxicab Metric
  9. Finite Fields
  10. Hamming Distance and Error Correcting Codes

Cryptology

  1. Secret Decoder Rings
  2. Transposition Ciphers
  3. Playing with Historical Ciphers (Applet)
  4. Letter Frequencies in English
  5. Breaking an Affine Cipher
  6. The Hill Cipher
  7. The Vigenere Cipher
  8. The Diffie-Hellman Key Exchange Protocol
  9. RSA and Public Key Codes

Mathematical Inquiry

  1. Good Questions to Ask
  2. An Investigation of Pascal's Triangle
  3. A Curious Inequality...
  4. Interesting Problems to Investigate

Practice

  1. Exercises - Divisibility
  2. Exercises - Arguing by Contradiction
  3. Exercises - The Pigeonhole Principle
  4. Exercises - Induction and Sums
  5. Exercises - Induction in Other Contexts
  6. Exercises - The Well Ordering Principle
  7. Exercises - Patterns and Conjectures
  8. Exercises - The Collatz Conjecture
  9. Exercises - Pythagorean Triples
  10. Exercises - The GCD and the Euclidean Algorithm
  11. Exercises - Permutation Puzzles and the GCD
  12. Exercises - Linear Combinations
  13. Exercises - Linear Congruences
  14. Exercises - The Fundamental Theorem of Arithmetic
  15. Exercises - Vectors and Linear Transformations
  16. Exercises - The Hill Cipher
  17. Exercises - The Vigenere's Cipher
  18. Exercises - Continued Fractions
  19. Exercises - Fast Exponentiation and Fermat's Little Theorem
  20. Exercises - Cardinality and Infinite Sets
  21. Exercises - RSA and Public Key Codes
  22. Exercises - The Order of an Integer and Primitive Roots
  23. Exercises - Index Arithmetic
  24. Exercises - Legendre's Symbol

Review

  1. Review Exercises (Set A)
  2. Review Exercises (Set B)
  3. Review Exercises (Set C)

For Fun

  1. UK codebreakers unable to decipher message from WWII carrier pigeon
  2. Reddit Unlocks the Secret of Cryptic Subway Code
  3. Help the FBI crack a secret code (and maybe help them solve a murder).
  4. The Kryptos Sculpture
  5. The Voynich Manuscript