Syllabus

Divisibility: Euclid’s division lemma, Divisibility, Linear Diophantine equations, Combinatorial and computational number theory: Fermat’s little theorem, Wilson’s theorem, Generating functions, use of computers in number theory. Fundamentals of congruences: Basic properties of congruences, Residue systems, Linear congruences, Theorems of Fermat and Wilson revisited, Chinese remainder theorem, Polynomial congruences, Arithmetic functions: Combinatorial study of j(n), Formulae for d(n) and s(n), Multiplicative arithmetic functions, Mobius inversion formula, Primitive roots: Properties of reduced residue systems, Primitive root modulo p. Quadratic congruences: Quadratic residues, Legendre symbol, Quadratic reciprocity, Jacobi symbol, Pythagorean triangles. Special Nonlinear Diophantine equations: Expression of numbers as sum of squares, Pell’s equation.

To motivate for research in number theory

To review basic number theory

To make the students acquainted with quadratic Diophantine equations

To develop creative thinking

The students will be motivated for research in number theory.

G.E. Andrews, Number Theory, Dove Publication Inc , 1994

Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag , 1976

Thomas Koshy, Elementary Number Theory with Applications, Elsivier , 2008

G. H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Oxford University Press , 2008