Here’s an interesting little mathematical morsel from the pages of the bit-player blog having to do with two topics I’ve found interesting in the past: prime numbers and random walks.
Let’s consider the sequence of prime numbers > 3. All such primes are congruent to -1 or to 1 modulo 6. So, let’s use that as the “random” variable controlling a random walk. If you consider all odd integers, you can generate a walk as follows. Take a step in the current direction. If the number is composite, you are finished. Otherwise it is a prime. If it is congruent to -1 modulo six, then turn left, otherwise turn right. You end up with a “random” walk, with several interesting questions about whether it shares properties with real “random” walks. Check the blog entry for more discussion:
I can visualize an interesting program written to draw these which is a modification to the segmented sieve algorithm that I’ve coded up previously. Each “segment” generates a segment of the overall path, and as long as you know the coordinates of the starting position, you can overlay and merge these points with reasonable efficiency. I might have to give that a go some time.
Addendum: In searching for more material by Gruenberger, I discovered that he was an early proponent of the educational uses of computing, and that some of his papers are available for download from the RAND corporation.