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