E-learning in Mathematics at Undergraduate and Postgraduate Level

Syllabus:

1. Revision :- Divisibility in integers, Division algorithm, G.C.D., L.C.M.. Fundamental theorem of arithmetic. The number of primes. Mersenne numbers and Fermat numbers.

2. Congruences :- Properties of congruence relation. Residue classes and  their properties. Fermat's and Euler's theorems. Wilson's Theorem. The congruence X^2 = -1 (mod p) has   solution if and only if p is the form 4n+1 where p is a prime. Linear congruences. Chinese Remainder Theorem.

3. Techniques of numerical computation, Public-Key Cryptography,Congruences with Prime Power Moduli, Prime Modulus, Primitive roots and power residues for a prime modulus.

4. Arithmetic functions : Euler function, Greatest integer function, Divisor function d(n), Moebius function µ(n). Properties and interrelation.

5. Quadratic Reciprocity :- Quadratic residues, Legendre symbol and its properties, Quadratic reciprocity law, Jacobi symbol and its properties. Sums of Two Squares.

6. Some Diophantine Equations: The equation ax + by = c , simultaneous linear equations.

7. Algebraic Numbers :- Algebraic Numbers, Algebraic number fields. Algebraic integers, Quadratic fields. Units in Quadratic fields. Primes in Quadratic fields. Unique factorization, Primes in quadratic fields having the unique factorization property.

Text Book:

• Ivan Niven & H.S. Zuckerman, An Introduction to Number Theory (Wiley Eastern Limited) 

• Sections: 2.1 to 2.8, 3.1 to 3.3, 3.6, 4.1 to 4.4, 5.1, 5.2, and 9.1 to 9.9.

Reference Books:

1. T. M. Apostol, An Introduction to Analytic Number Theory (Springer International Student's Edition) 

2. David M. Burton, Elementary Number Theory (Universal Book Stall, New Delhi) 

3. S. G. Telang, Number Theory (Tata Mcgraw-Hill) G. H. Hardy and E. M. Wright, Introduction to Number Theory (The English Language Book Society and Oxford University Press)