|Course Archives Theoretical Statistics and Mathematics Unit|
Course: Number Theory
Time: Currently not offered
| Syllabus |
Syllabus: 1. Review of unique factorization; properties of the rings Z[i] and Z[?] (chapter 1 of IR).
2. Review of congruences, Euler's f-function, results of Fermat, Euler and Wilson; linear congruences, Chinese remainder theorem. Primitive roots and the group structure of U(Z/nZ); applications to congruences of higher degree; Hensels Lemma (chapter 4 of IR and sections 2.1 to 2.7 of NZM).
3. Quadratic Reciprocity: Quadratic Residues, Gaussian reciprocity law, the Jacobi symbol (chapter 5 of IR).
4. Arithmetic Functions, Moebius inversion formula and combinatorial methods like principle of inclusion-exclusion and pigeonhole etc (sections 4.2,4.3,4.5 of NZM).
5. Diophantine equations. Linear equations, the equation x^2+y^2=z^2. Method of Descent; the equation x^4+y^4=z^2 (section 5.1 to 5.4 of NZM).
6. Binary Quadratic forms. Sum of two squares. Legendre's Theorem (section 3.4 to 3.7 of NZM).
7. Simple continued fractions. Infinite continued fractions and irrational numbers. Periodic continued fractions, algorithms for solving Brahmagupta-Pell equation, numerical computations. Dirichlet's box principle and solution of Pell's equation (chapter 7 of NZM).
8. Elementary results on the function p(x), Bertrands postulate (sections 8.1, 8.2 of NZM).
Suggested Texts :
1. (IR) K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory Second Edition, Springer (Indian reprint 2004).
2. (NZM) I. Niven, H.S. Zuckerman and H. Montgomery, An Introduction to the Theory of Numbers, John Wiley (1991); Indian edition available.
3. J.H. Silverman, A Friendly Introduction to Number Theory, Prentice-Hall (2005).
4. J. Stillwell, Mathematics and Its History Second Edition, Springer (Indian reprint 2004).
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