PMAT 32342: Number Theory

Course Code : PMAT 32342

Title               : Number Theory

Pre-requisites   :  PMAT 12242

Learning Outcomes:

After the completion of this course unit, the student will be able to:
1. demonstrate a foundational understanding of number theory, including the definitions, conjectures, and
theorems that permit exploration of topics in the field.
2. work with numbers and polynomials modulo a prime, linear congruences, and systems of linear
congruences, including their solution via the Chinese remainder theorem.
3. Apply mathematical ideas and concepts within the context of number theory.
4. Solve a range of problems in number theory.
5. demonstrate an understanding of the mathematical underpinning of cryptography.

 Course Contents:

Divisibility: 

• Division and Linear Diophantine Equations, Greatest common divisor, Euclidean Algorithm, Prime and Composite Numbers, Fibonacci and Lucas Numbers.

Congruences: 

• Linear Congruences, Chinese remainder theorem, systems of linear congruences, Applications of Congruences: Divisibility tests, Round-robin tournament, Special Congruences: Wilson’s theorem, Fermat’s little theorem, Euler’s theorem.

Multiplicative Functions: 

• Euler’s phi-functions.

Primitive Roots: 

• order of an integer and primitive roots, Primitive roots for primes, Primality testing using primitive roots, Perfect Numbers.

Cryptology: 

• Affine Ciphers, Hill Ciphers, Exponentiation Ciphers.

Quadratic Residues and Reciprocity: 

• Quadratic Residues, The Legendre Symbols, Quadratic Reciprocity.

 Method of Teaching and Learning : A combination of lectures, tutorial discussions and presentations.

 Assessment : Based on tutorials, tests, presentations and end of course examination.

 Recommended Reading:

1. K. Rosen, Elementary Number Theory and its Applications (6th Edition), Pearson (2010).
2. T. Koshy, Elementary Number Theory with Applications (2 nd Edition),‎ Academic Press (2007)
3. G. Andrews, Number Theory, Dover Publications (1994)
4. O. Ore, Number Theory and Its History, Dover Publications (1988)
5. W. Trappe, L. Washington, Introduction to Cryptography with Coding Theory (2 nd Edition), Pearson
International Press (2006)