MTH202: Elementary Number Theory
MTH202: Elementary Number Theory
The course is mandatory for all pre-majors chosing mathematics as one of the disciplines.
The course is also available as an IDC to any premajor not chosing mathematics as a discipline.
If anyone intends to audit the course, please drop me a mail.
Classes Monday and Tuesday 10:00am to 11:00am LH5
Reserved Slot: Thursday 12:00am to 1:00pm LH5
Office hours are by appointment. Best time to catch: immediately after the class. Location: AB2-1F1
Grading Policy: Midsem I - 20, Midsem II - 20, Endsem - 50 to 60, Quizes/Assignments/Attendance : 0 to 10.
Links: (to appear over the course of semester)
We will broadly follow the official course outline and the references below.
However we will take liberty to modify the same, subject to our interests and pace.
Primary reference for this class will be the lectures themselves.
Official Course Outline
The division algorithm and the the greatest common divisor algorithm. Chinese remainder theorem as an application.
Modular arithmetic. Linear congruences.
Prime numbers and irreducible numbers. Divisibility modulo a prime.
Principle of mathematical induction and the proof of unique factorisation of a number into prime factors.
Infinitude of primes. Euclid’s and Euler’s proofs.
The sieve of Eratosthenes and the expected distribution of primes.
Fermat’s little theorem and introduction to primality testing.
Euler’s totient function and the multiplicative group modulo an integer. Applications to decimal and other expansions. Euler’s generalization of Fermat’s little theorem that uses the totient function.
Arithmetical functions. Multiplicative functions (totient, divisor, etc.) Moebius inversion formula and some elementary applications.
Additional Topics.
Euler’s criterion for quadratic residues, and a discussion (without proofs) of Legendre symbol and the reciprocity law.
Farey fractions, Stern-Brocot tree and continued fractions. Relation to Fibonacci sequence.
Sums of powers, binomial coefficients, exponential growth and Fibonacci sequence.
Official References
Elementary Number Theory, D. Burton, (7th Edition), McGrawHill (2017).
Number Theory, G. H. Hardy and E. M. Wright, (6th Edition), Oxford University Press (2008).
Concrete Mathematics, R. Graham and D. Knuth, Addison-Wesley (1989).