Anyone have a favorite recreational math tidbit that involves primitive roots? Three that I know about are card shuffling, radix expansions of fractions, and the Ducci process. (Shuffle a deck of 2n cards with successive perfect inshuffles. It might take 2n-1 iterations before the original order is restored; for what n does it happen? For what n does the binary expansion of 1/n repeat with period n-1? The Ducci process is described at https://en.m.wikipedia.org/wiki/Ducci_sequence ; for what n is it the case that the length of the eventual repeating cycle depends only on n and not on the specific choice of not-all-equal integers written at the vertices of the original n-gon? The answers to all three questions involve the issue of which primes p have 2 as a primitive root.) Jim Propp