Let’s quickly recap what happened last time. We started with two coprime numbers, p and q, and for every number n which was neither a multiple of p nor q we wrote n as

n=ap + bq

where 0<a<q. We defined the function j(n) as +1 if b was positive, and -1 if b was negative (and zero if n was not an allowed number). The function s(n) was then defined as the running total of the j’s; for example, s(3) = j(1)+j(2)+j(3).

The question was then: how do we predict the function s? The graphs we saw in the last post showed that s had all sorts of bumps and wiggles that varied a lot for different values of p and q. How should we go about discovering the pattern?

Let’s try condensing all the data of s into just one number. Then we can compare the value of this number over lots of values of p and q. The number I’m going to choose is ‘the area under the graph of s’. That’ll be the area inside the ‘V’ shape of the graph, since s is always negative. It is easy to set up a computer program to calculate this value for lots of different p and q less than 100. Here is the result:

Value of the area under the graph of s for different values of p and q. Red=small...Bluer=bigger.

Don’t worry about the dark spots in the picture: those are the points when p and q weren’t coprime, so there was no graph to find the area of. The exciting thing is the very regular colouring of the graph! The values of the area are changing very predictably with p and q, despite the graphs themselves being very unpredictable.

What is this new ‘area’ function? What is it really measuring and how can we find a formula for it? Tune in next time for a deeper explanation of what is going on here…

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