Specific Objectives of course:

The focus of the course is on study of the fundamental properties of integers and develops ability to prove basic theorems. The specific objectives include study of division algorithm, prime numbers and their distributions, Diophantine equations, and the theory of congruences.

Course Outline:



Well-ordering principle. Principle of finite induction.

Divisibility theory:

The division algorithms. Basis representation theorem. Prime and composite numbers. Canonical decomposition. The greatest common divisor. The Euclidean algorithm. The fundamental theorem of arithmetic. Least common multiple.

Linear Diophantine equations:

Congruences.  Linear congruences. System of linear congruences. The Chinese remainder theorem. Divisibility  tests.  Solving  polynomial  congruences.  Fermat’s  and Euler’s theorems. Wilson’s theorem.

Arithmetic functions:

Euler’s phi-function.   The functions of J and sigma.  The  Mobius  function.  The  sieve  of  Eratosthenes.  Perfect numbers. Fermat and Mersenne primes.

Primitive  Roots  and  Indices:

The  order  of  an  integer  mod  n. Primitive roots for primes. Composite numbers having primitive roots.

Quadratic  residues:

Legendre  symbols  and  its  properties.  The quadratic reciprocity law. Quadratic congruences with composite moduli. Pythagorean triples. Representing numbers as sum of two squares.