MAS2202 - Introduction of Number Theory

Topical Outline with Specific Course Objectives

1. Linear and Nonlinear Diophantine Equations
   1. Solve Linear Diophantine equation in two variables.
   2. Represent a Primitive Pythagorean Triples with a unique pair of relatively prime integers.
   3. Investigate the historical background of Fermat's Last Theorem.
2. Primes and Greatest Common Divisions
   1. Investigate the distribution of prime numbers.
   2. Represent integers in different bases.
   3. Find the greatest common factor using the Euclidean Algorithm.
   4. Investigate different factorization methods, such as the sieve of Eratosthenes and Fermat factorization.
   5. Investigate the proof of the Fundamental Theorem of Arithmetic.
3. Congruence
   1. Solve systems of linear congruences.
   2. Solve systems of linear congruences with different moduli using the Chinese Remainder Theorem.
   3. Be able to factor using the Pollard Rho Method.
   4. Use Wilson's Theorem and Fermat's Little Theorem as the basis for primality tests and factoring algorithms.
   5. Investigate Pseudo-primes.
   6. Investigate Carmichael numbers.
   7. Develop divisibility tests.
   8. Describe how congruences are used to detect errors in strings of digits.
4. Multiplicative Functions
   1. Determine if a function is multiplicative using the Euler Phi-function.
   2. Find the value of the Euler-Phi function for integers.
   3. Investigate perfect numbers and Mersenne prime numbers and their connection.
   4. Explore the use of arithmetical functions, the Mobius function, and the Euler totient function.
   5. Investigate the Dirichlet product of arithmetical functions.
5. Cryptography
   1. Learn to encrypt and decrypt a message using character ciphers.
   2. Learn to encrypt and decrypt a message using Public-Key cryptology.