A weird mystery is almost solved.

A project log for PEAC Pisano with End-Around Carry algorithm

Add X to Y and Y to X, says the song. And carry on.

Yann Guidon / YGDESYann Guidon / YGDES 03/12/2023 at 20:470 Comments

PEAC is striking because it looks "too simple to be good". One of the surprising features is it does not rely on a prime number to inject some irrationality in a mathematical development. In particular, PEAC's statespace's size has a formula of (N×(N-1))-2 which is always an even number for any integer N. This evenness makes the size impossible to be a prime number so where does the maximal length come from ?

Well it comes from something different that I just uncovered. The PEAC generator is actually a sort of filter that is looped on itself. This filter has some interesting properties and works "best" when the input range is one more than the output range. It does not depend on primality "as such" but on relative primality, and since N and N-1 can't have common prime factors, it works (for certain values of it).

It has come together nicely in my head but it will take time to be explained easily. At least now I'm not mystified anymore :-)