Sums of Squares - Peter Giblin
24th November 2012
How can a rectangle be filled exactly with squares?
The problem of doing this with squares all of different sizes is very difficult but I will allow some squares to have the same size, and describe some solutions.
One method will lead to a famous calculation called the Euclidean algorithm, and to continued fractions which are very important in applications of number theory to, for example, factoring problems. I'll also touch on the subject of expressing whole numbers as sums of squares, something which is of course much easier (is it?) than filling rectangles with squares. For example, no number which leaves remainder 3 when divided by 4 can be a sum of two squares.