I want to tell you about the current mathematics I’m working on, because it’s exciting, surprising, beautiful, deep, and easy to explain! Everything that a good piece of maths should be. I’m going to have to explain it in a few parts though, leading you through the different steps I had to solve to get to the solution.
Here’s a problem to get you started. I’m going to give you two numbers, p and q, which are coprime. That means they have no common factors. For example, I can’t give you the numbers 9 and 12, since both of them are divisible by 3. But I can give you the numbers 9 and 10 since 9=3×3 and 10=2×5, and there is no overlap. Ok. Now I give you another number n between 1 and pq-1, such that n is neither a multiple of p nor a multiple of q. Your task: take n and keep minusing multiples of p from it until you get a number that is a multiple of q. Then tell me whether the multiple of q is positive or negative.
Let’s do an example. Suppose p=3 and q=5. I need to give you a number n between 1 and 14, so let’s start with the simple n=1. First we try 1-3=-2. Not a multiple of 5. So we try 1-(2×3) = -5. Success! And -5 is negative. Now let n=8. Try 8-3=5. First time lucky! And this time we got a positive number.
For particular numbers p and q, we are going to do this for all possible n. So if p=3 and q=5, we have to find the answer for n=1,2,4,7,8,11,13,14. If the answer is positive, put j(n)=1, and if it’s negative we put j(n)=-1. So, for these particular values of n, the corresponding values for j are -1,-1,-1,-1,1,1,1,1. Now we keep a running total, and call that number s. So the values for s are -1,-2,-3,-4,3,2,1,0. Actually, for reasons that I will explain later, the interesting number to look at is 2 times s.
Let’s draw a graph of the function 2s and see what it looks like. Does it always go “minus minus minus plus plus plus”? Well, for p=2, the answer is ‘yes’, no matter what value you choose for q. Here’s the graph of p=2,q=25, with the values of n running along the bottom and the value of 2s running up the side.
Notice the lovely ‘V’ shape that it makes. The values of s decrease until a certain point, and then increase again.
Let’s see what happens when p=3. Here’s the graph for p=3, q=10:
Oh no, what happened? The graph is all wiggly at the bottom! Minus, plus, minus, plus – what’s going on? It’s obviously not a proplem with the number 3, since our previous example of p=3, q=5 worked out fine. Maybe we just picked a funny combination of numbers. We had better try another pair and see what happens there. How about p=7, q=16?
Even more wiggles! Yikes, it seems that this function is going to be pretty unpredictable. The pattern seems to be very different for every p and q. How on earth are we going to get a formula that tells us what is happening?
Tune in next time for more clues…