Graphs

Included in the appendix are three graphs that were created in Matlab, using the output from my programs, as well as other data that was available for base 10 PDIs and PPDIs of orders up to 100 ([11]) . These graphs demonstrate the sporadic nature of PDIs and PPDIs, which made it impossible (for me) to calculate a good bound.

The first of these graphs contains three plots - the number of PDIs versus order, the number of PPDIs versus order, and the number of PDIs plus the number of PPDIs versus order. One interesting thing to notice about this graph is that no single order has more than 9 PDIs or PPDIs. However, looking at other bases, it does not appear that any similar statement can be made - for instance, in base 4, there are 6 order-3 PPDIs. Another thing to notice is the irregularity with which they occur. This does not appear to be a well-behaved function, and in fact is very sporadic. This is one of the main reasons that I was unable to create a bound for it.

The second graph contains an overlaid plot of PDIs and PPDIs versus order. It is interesting to notice that there are similarities in the peaks and valleys of the two graphs, although generally they don't follow each other. Beyond order 43, there are no PPDIs in base 10, so there is definitely no resemblance between the two graphs from that point on.

The third graph shows an attempt to smooth out the function that represents the number of PDIs and PPDIs versus order. The dotted line in this graph is the number of PDIs and PPDIs, and the solid line represents the midpoint between two consecutive values on the graph. I did this to see if there some smoother graph that perhaps the function was oscillating around, but as can be seen, this function is still quite sporadic as well.