Course Summary

Maths - Number Theory

Topic 1
Foundations, Twin Primes and Prime Decades, Arithmetic Progression of Primes, Ulam's Spiral

Topic 2
This Unit introduces Triangular Numbers and then Polygonal Numbers and various formulae for the number of dots and relations between the numbers.  Runsums - the sums of consecutive numbers - are also covered.

Topic 3
This Unit commences with expressing numbers as the sums of squares and the main theorems associated with that. It then proceeds to looking at numbers as the sums of polygonal numbers; and follows with Waring’s Problem on expressing numbers as sums of cubes and higher powers

Topic 4
The Unit covers two famous sets of numbers which have fascinated amateur and professional mathematicians over the centuries.

Topic 5
Modular Arithmetic is introduced including calculating multiplicative inverses and using Fermat's Little Theorem.  This is followed by the application of modular arithmetic in encryption explaining  Affine, Exponential and RSA Cyphers.

Topic 6
A Diophantine Equation is a polynomial equation with two or more variables for which only integer solutions are sought. The solutions may contain negative as well as positive integers. In Unit 4 we came across an example: x2 + y2 = z2 with Pythagorean Triples as solutions. This Unit will cover linear and quadratic Diophantine Equations and some higher order ones.