Number theory
4 points * First semester * Clayton * Prerequisites: MAT2020
Objectives On the completion of this subject students will be able to compute the greatest common divisor of two numbers and establish some of its basic properties; demonstrate the existence of an infinite collection of primes and establish some basic results concerning prime numbers; solve congruences and apply congruence theory to the solution of Diophantine equations; calculate residue systems, establish Euler's Theorem and Wilson's Theorem; solve simultaneous linear congruences and polynomial congruences; establish properties of quadratic residues and develop a familiarity with the Legendre symbol and Jacobi symbol; prove Gauss's Law for quadratic reciprocity, and applications to the solution of quadratic congruences; calculate continued fraction expansions and compute rational approximations to irrationals.
Synopsis Primes, unique factorisation and greatest common divisor. Modular arithmetic, quadratic residues and primality testing. Continued fractions and approximation by rationals. Public key cryptosystems.
Assessment Examinations (1.5 hours): 70% * Assignments: 30%
Recommended texts
Niven I, Zuckerman H S and Montgomery H The theory of numbers Wiley, 1991
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