Weird Science: The Curious Divisions of Dr Mike Hirschhorn!

Curious divisions = weird and beautiful mathematics!!

Certain numbers can be broken down into different squares, for example

25 = 5^2 and

25 = 4^2 + 3^2.

Thus there are exactly two different ways of expressing the number 25 via a sum of different squares.

How about 100?

Well,

100 = 10^2 and

100 = 8^2 + 6^2 and

100 = 7^2 + 5^2 + 4^2 + 3^2 + 1^2

so there are exactly three different ways of expressing the number 100 via a sum of different squares.

If we let p denote the number of "partitions" of the number n into different squares, then for the above examples p(25) = 2 and p(100) = 3.

Similarly, p(1000) = 1269.

If we divide the general function p(n) by a very special exponential-type function (which we call "the H function" in honour of Mike!) then we get these very beautiful graphs!

Mike is currently trying to explain these awesome curves.