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Durham University

Department of Mathematical Sciences

MATH2617 Elementary Number Theory II

This course introduces the fundamental methods and techniques used in elementary number theory together with some of their applications. It starts with the discussion of the basic properties of the integers, such as the Euclidean algorithm and the fundamental theorem of arithmetic, that is, the unique factorisation into prime numbers. Then we discuss congruences and modular arithmetic culminating in the classic quadratic reciprocity law which relates the question when a prime p is a square modulo another prime q to the reciprocal question when q is a square modulo p. This problem and the search for generalisations have been driving number theory for the last 200 years. In the next part of the course we consider some classical Diophantine problems such as to find integer solutions for Pythagorean equation x²+y²=z², to represent positive integer numbers as the sum of squares or to find integral solutions for Pell's equation x²-dy²=±1 for d a square-free positive integer. Pell's equation in turn leads us to the world of continued fractions which play an important role in modern mathematics (particularly in the area called Diophantine approximation).

Outline of Course

Aim: To introduce fundamental topics in elementary number theory.

  • GCD, prime and composite numbers, factorisation of numbers, fundamental theorem of arithmetic (3 lectures)
  • Residues, Euler's quotient function, Euler-Fermat theorems, Chinese Remainder theorem. (4 lectures)
  • Primitive roots and indices. (2 lectures)
  • Quadratic residues and quadratic reciprocity law. (3 lectures)
  • Pythagorean triples and Fermat equation x⁴+y⁴=z⁴. Method of Fermat descent. (2 lectures)
  • Representation of a number as a sum of squares. Fermat and Lagrange theorems. (3 lectures)
  • Continued fractions and Pell's equation. (3 lectures)

Prerequisites

For details of prerequisites, corequisites, excluded combinations, teaching methods, and assessment details, please see the Faculty Handbook.

Reading List

Please see the Library Catalogue for the MATH2617 reading list.

Examination Information

For information about use of calculators and dictionaries in exams please see the Examination Information page in the Degree Programme Handbook.