Course Summary

Maths - Number Theory Course Leader

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

Unit 2
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 and an asterisk marks a section which may a bit more difficult to follow.

Unit 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.

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

Unit 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.

Unit 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. 

Unit 7
Some prime number topics - the Fermat, Mersenne, and Sophie Germain Primes; a formula that generates primes; and the Prime Number Theorem and the Riemann Hypothesis.

Additional week to continue with Unit 7