Number Theory-PMATD30211 

Unit 1: Division algorithm, Euclid's algorithm, prime numbers, fundamental theorem of arithmetic, distribution of primes, discussion of the Prime Number Theorem, the series of Reciprocals of primes, congruences, Goldbach conjecture, Twin-prime conjecture, Linear Congruence, Chinese remainder theorem, Fermat’s little theorem, Wilson’s theorem, Euler’s theorem. Applications: Public key encryption, RSA encryption, and decryption with applications in security systems. 

 Unit 2: Elementary arithmetical functions, perfect numbers, Mersenne primes and Fermat numbers, Irrational numbers-Irrationality of mth root of N, e, and π. 

Unit 3: Primitive roots and indices, Quadratic residues, Legendre symbol, Gauss’s Lemma, Quadratic reciprocity law, Jacobi symbol. 

Unit 4: Fermat's two square theorems, Lagrange's four square theorem, Diophantine equations: ax + by = c, x^2 + y^2 = z^2, x^4 + y^4 = z^2, sums of two and four squares. 

Reference: 

• D.M. Burton (2010), Elementary Number Theory, 7th Edition. McGraw-Hill Education. 

• G.H. Hardy and E.M. Wright (1975), An introduction to the Theory of Numbers, 4th Edition. Oxford University Press. 

• I. Niven, H. S. Zuckerman and H. L. Montgomery (2004), An Introduction to the Theory of Numbers, New York, John Wiley and Sons, Inc., 5thEd.

• T. M. Apostol (1998), Introduction to Analytic Number Theory, Narosa Publishing House, New Delhi.