Courses

Theorems and conjectures on prime numbers: primes in arithmetic progressions, primes of special form, formulas that give prime numbers, distribution of primes. Arithmetic functions: number of divisors, sum of divisors, Euler function, Möbius function, Dirichlet convolution, Möbius inversion formula. Mersenne numbers, perfect numbers, Fermat numbers. Polynomial equations modulo n, initial roots modulo n, quadratic residues, Legendre symbol, Jacobi symbol, Kronecker symbol, quadratic inversion law. Pythagorean triples, nonlinear Diophantine equations, Fermat's method of infinite descent, Pell's equation. Continued fractions, properties of convergents, optimal approximations of rationals from rationals, periodicity of continued fractions. Dirichlet and Liouville theorems for Diophantine approximations, elements of transcendental Number Theory. Representation of an integer as a sum of squares or as a sum of higher powers, the Waring problem. Symmetric and non-symmetric cryptography. Pseudoprimes, Carmichael numbers, deterministic and non-deterministic prime verification algorithms. Integer factorization algorithms.