I have to confess that most often my eyes begin to glaze over when someone starts discussing the prime numbers. However, my ears have perked up at times over the primes--maybe first when I learned of their use by Godel and second, of their links to the Riemann zeta function and related analysis. More recently I have encountered the primes when observing the simplicity/regularity of indexed expressions in certain sequences of expressions when the index is prime in contrast to those for the non-prime indices, such as for the cyclotomic polynomials (see "A tangential note" in this MO-A) and for certain colored necklaces (see this MO-Q). In the interest of stimulating interest in or a more informed appreciation of the primes for those who are not classical number theorists, I'd like to ask
In what scenarios do the primes occur that you find of particular interest or utility outside of pure number theory?
Anecdote: Once long ago I was quickly passing through a wing of the Getty Art Museum, the Roman villa in the Palisades in L.A.--a wing containing medieval art with its religious motifs, at that time of little interest to me. I stopped in front of a docent who was enthusiastically informing a group of the symbology underlying a painting. She posed questions that revealed the painting subtly and cleverly indicates three events in the life of Christ in what appears on the surface to be a single moment frozen in time. I now appreciate the depth of meaning in such art.