Course Details
Subject {L-T-P / C} : MA6623 : Advanced Number Theory { 3-1-0 / 4}
Subject Nature : Theory
Coordinator : Prof. Gopal Krishna Panda
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.
Course Objectives
- 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
Course Outcomes
The students will be motivated for research in number theory.
Essential Reading
- G.E. Andrews, Number Theory, Dove Publication Inc , 1994
- Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag , 1976
Supplementary Reading
- 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