Continuing with the previous post, here are some graphics that illustrate what I was pointing to. Interestingly, since the file I did at home was over 4 megs in size, I couldn’t save it to bring it to work to do the graphics. However, I discovered using the MOD math trick that you can easily create one of these diagrams for any value that your calculator will do the MOD function on :-) Simply pick a base-line value (in this case, I used the lowest cell possible in Excel, at 65536). Calculating 1 and 2 is easy (1 is all numbers, 2 is the even numbers). With three, take your baseline number and find the baseline MOD 3 (e.g. 65536 MOD 3 = 1). Move up a number of cells equal to the value you got in your MOD calculation. Then, repeat the series (for three, you would simply count up three each time). With that method, you can easily calculate a diagram like the following using very little time–in fact, I did it in less than half of my lunch break. As with the other diagrams, click on the graphic to see a larger version of it:
Here you can see the “spike” I mentioned at row 65520. Furthermore, you can see that there are some numbers that shoot off at a consistent angle, namely:
But there are some gaps in the branches. Interestingly enough, those gaps correspond to the gaps in the 65520 line:
In any case, these graphics should help you to understand a bit more what I was referring to in the previous post, when I referred to a spike at 65520. Notice that there are no values in the range selected at 65519 or 65521. My hypothesis is that either these numbers are prime, or the values that would finally hit on those lines are going to be prime numbers.
