Course Syllabus
This syllabus outlines the modules of Number Theory.
- Divisibility
- Euclidean algorithm
- greatest common divisor
- congruences; residue classes
- complete and reduced residue systems
- Fermat theorem; Euler theorem; Wilson theorem
- Chinese remainder theorem and applications
- Arithmetic functions; divisor function; sum of divisors function
- Euler phi function
- Mobius function; Mobius inversion formula
- multiplicative functions
- convolution of arithmetic functions
- applications to counting problems
- Linear Diophantine equations
- Pythagorean triples
- quadratic Diophantine equations: selected examples
- Pell equation: introduction
- representation of integers as sums of squares: elementary discussion
- continued fractions and rational approximations: introduction
- Order of an integer modulo n
- primitive roots; existence of primitive roots: selected cases
- quadratic residues; Legendre symbol; Euler criterion; Gauss lemma
- law of quadratic reciprocity: statement and applications
- Classical cryptography: types and examples
- public-key cryptography: elementary idea
- RSA cryptosystem