Course Archives Theoretical Statistics and Mathematics Unit | ||||||||
Course: Number Theory Instructor: Ramdin Mawia Room: G25 Level: Postgraduate Time: Currently offered |
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Syllabus Past Exams Syllabus: i) Brief review of Division Algorithm, gcd and lcm, Euclidean algorithm; Linear Diophantine equations, congruences and residues, the Chinese Remainder Theorem; The ring Z=nZ and its group of units, The Euler -function, Fermats little theorem, Eulers theorem, Wilsons Theorem, Sums of two and four squares. ii) Pythagorean triplets and their geometric interpretation (rational points on circles); Rational points on general conics; Fermats method of infinite descent and application to simple Diophantine equations like x4 + y4 = z2; The Hasse principle for conics (statement only), Brief discussion on rational points on cubics and the failure of the Hasse principle (statement only). iii) Polynomial congruences and Hensels Lemma; Quadratic residues and nonresidues, Eulers criterion, Detailed study of the structure of the group of units of Z=nZ, Primitive roots; Definition and properties of the Legendre symbol, Evaluation of Gauss sums, The law of quadratic reciprocity for Legendre symbols; Extension to the Jacobi symbols. iv) Arithmetical functions and their convolutions, multiplicative and completely multiplicative functions, examples like the divisor function d(n), the Euler function (n), the Mobius function (n) etc.; The Mobius inversion formula; Sieve of Eratosthenes; Notion of order of magnitude and asymptotic formulae; Euler and Abel summation formulae, Hyperbola method of Dirichlet, Average order of magnitude of various arithmetical functions such as (n); d(n) etc.; Statement of the Prime Number Theorem; Elementary estimates due to Chebyshev and Mertens on primes. v) Algebraic integers, Arithmetic in Z[i] and Z[!], Primes of the forms x2 + y2 and x2 + xy + y2; Integers in quadratic number fields; Examples of failure of unique factorization; Units in the ring of integers of a real quadratic field and application to the Brahmagupta-Pell equation. vi) One or more topics from the following list can be discussed if time permits: a) Absolute values on Q, Ostrowskis Theorem, Completions of Q, Qp and Zp, The p-adic topology. b) The notion of algebraic and transcendental numbers; Transcendence of e, Diophantine Approximation, Dirichlets Theorem; Liouvilles Theorem, Statement of Roths Theorem. c) Continued fractions; Applications to Diophantine approximation, Application to the Brahmagupta-Pell equation. d) Introduction to Geometry of numbers. e)Application of Number Theory to RSA and other cryptosystems. Suggested Texts : (a) T. Apostol, Analytic Number Theory. (b) D. Burton, Elementary Number Theory. (c) G.H Hardy and E.M. Wright, An Introduction to The Theory of Numbers. (d) K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory. (e) Manin and Panchishkin, Introduction to Modern Number Theory. (f) S.J. Miller and R. Takloo-Bighash, An Invitation to Modern Number Theory. (g) I. Niven, H. S. Zuckerman, H. L. Montgomery, The Theory of Numbers. (h) J.-P. Serre, A Course in Arithmetic. Evaluation:
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