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Indian Institute of Science Education and Research Bhopal
MTH 305: Elementary Number Theory (4)
Pre-requisites (recommended): MTH 101: Calculus of One Variable
Learning Objectives:
The aim of this course is to develop a conceptual understanding of the elementary theory of numbers and to expose the students to writing proper mathematical proofs.
Course Contents:
Foundations: Principle of mathematical induction (with emphasis on writing a few basic proofs), binomial theorem, countable and uncountable sets, some basic results on countability, countability of Z, Q and uncountability of R.
Divisibility: Basic properties, division algorithm, GCD, LCM, properties of GCD, relation between GCD and LCM, Euclidean algorithm for finding GCD, Pythagorean triples, linear Diophantine equations, fundamental theorem of arithmetic, Euclid's lemma, existence of infinitely many primes.
Modular arithmetic: Basic properties of congruences, linear congruences, Chinese remainder theorem, Fermat's little theorem, Wilson's theorem.
Number theoretic functions: Arithmetic functions (tau, sigma and Mobius) and their properties (specifically multiplicative property of the functions tau, sigma and the Mobius inversion formula), Euler's phi function and its properties, Euler's Theorem, Fermat's little theorem as a corollary of Euler's theorem.
Quadratic reciprocity: Primitive roots (order of an integer modulo n, primitive roots for primes), quadratic congruences, definition of quadratic residue, Legendre symbol and its properties, quadratic reciprocity law.
Continued fractions: Finite continued fractions, approximation of rational numbers by finite simple continued fractions, solution of linear Diophantine equations using finite continued fractions, infinite continued fractions, unique representation of irrationals as an infinite continued fraction, Pell's equation and its solutions using continued fractions.
Suggested Books:
Textbooks:
- David Burton, Elementary Number Theory, 7th edition, McGraw Hill Education, 2012.
- John Stillwell, Elements of Number Theory, 1st edition, Springer, 2003.
References:
- James Tattersall, Elementary Number Theory in Nine Chapters, 1st edition, Cambridge University Press, 1999.
- Ya. Khinchin, Continued Fractions, 3rd edition, Dover, 1997.
- Thomas Koshy, Elementary Number Theory with Applications, 2nd edition, Elsevier, 2007.
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