A Basic Course in Number Theory

Overview

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This course intends to develop the basics of number theory touching upon many essential points such as the prime number theorem, quadratic reciprocity laws, Gauss’ theorem on the classification of binary quadratic forms, Brahmagupta-Pell equations, to quote a few. This course will enable a student to learn more advanced topics in number theory.
INTENDED AUDIENCE :Students with basic knowledge of Mathematics can take this course.
PREREQUISITES :Knowledge of basic group theory and ring theory will be useful but it is not necessary.
INDUSTRIES SUPPORT :None

Syllabus

COURSE LAYOUT

Week 1:Factorization of numbers, primesWeek 2:GCD, Euclid’s algorithm, properties of primesWeek 3:Arithmetical functions, examplesWeek 4:Dirichlet product, Möbius inversion
Week 5:Congruences, Chinese remainder theorem, primitive rootsWeek 6:Quadratic reciprocity law, applicationsWeek 7:Binary quadratic forms, Gauss’ theory of reduced formsWeek 8:Sums of two squares, sums of four squares
Week 9:Diophantine approximation, theorems of Dirichlet and LiouvilleWeek 10:Continued fractions, quadratic irrationalsWeek 11:Quadratic extensions of rationals, units in the rings of integersWeek 12:Diophantine equations with special reference to Brahmagupta-Pell equation