Number Theory
Number Theory (MAL7320)
January-April, 2020Lectures : Monday at 11:00 am, Wednesday at 10:00 am, Friday at 9:00 am
Discussion Hour : Thursday
Evaluation :
Mid Semester Examinations - 30% (15%+15%)
End Semester Examination - 30%
Quizzes - 30%
Assignments - 10%
Quizzes will be held on Jan 23, Feb 27, April 11 and April 24, 2020
Topics to be covered:
Introduction : Divisibility, prime numbers, properties and their distribution, complete and reduced residue systems, theorems of Fermat, Euler & Wilson, application to RSA cryptosystem.Congruences : Linear congruences, Chinese Remainder theorem, quadratic congruences, and Quadratic Reciprocity law.Arithmetical functions : Examples, properties and their rate of growth, Continued fractions, and their connections with Diophantine approximations, applications to linear and Pell’s equations, Binary quadratic forms, Partition: basic properties and results.Diophatine equations : Linear, quadratic and higher degree, some general equationsReferences:
David M. Burton, Elementary Number Theory, Mc Graw Hill Education, 2012.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press, 1960.
Niven, H. S. Zuckerman and H. L. Montgomery. An Introduction to the Theory of Numbers, Wiley, 1991
Course
3-January-2020 : Introduction and Motivation
6-January-2020 : Well ordering Principle, Mathematical Induction, Binomial Theorem
8-January-2020 : Divisibility
10-January-2020 : Division Algorithm, Euclid's Lemma
11-January-2020 : Prime numbers and their distribution
20-January-2020 : Congruences, complete residue system
21-January-2020 : Theory of linear congruences, Diophantine equations
22-January-2020 : Discussion on Assignment-I
23-January-2020 : Quiz-I
27-January-2020 : Chinese Remainder Theorem
28-January-2020 : Fermat's Theorem, Wilson's Theorem
29-January-2020 : Euler's Phi function
31-January-2020 : Euler's Theorem
1-February-2020 : Properties of Euler's Phi function
3-February-2020 : Discussion on Assignment-II
10-February-2020 : The sum and number of divisors
12-February-2020 : The Mobius Inversion formula
17-February-2020 : Quadratic Congruences
19-February-2020 : Primitive roots
26-February-2020 : Primitive roots of a prime number
28-February-2020 : Quiz-II
2-March-2020 : Quadratic Residue, Euler's Criterion
4-March-2020 : Legendre Symbol
6-March-2020 : Gauss Lemma
9-March-2020 : Quadratic reciprocity law
11-March-2020 : Quadratic Congruences with composite moduli
12-March-2020 : Quadratic Congruences with composite moduli (Contd.)
13-March-2020 : Introduction to Cryptography
14-March-2020 : Discussion on Assignment-IV
Classes shifted to online mode due to COVID