Course Archives Theoretical Statistics and Mathematics Unit
Course: Number Theory
Instructor: Ramdin Mawia
Room: G25
Time: Currently offered
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:
 Midterm Exam 30 marks Home Work / Assignment 20 marks Final Exam 50 marks Total 100 marks

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Past Exams
Midterm
 22.pdf 24.pdf
Semestral Supplementary and Back Paper
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