Please list some applications of prime numbers outside of
cryptography, factoring, hashing, and sorting.  I'm really looking for
physical things; examples would include ball bearings and gears
(minimizing cyclical wear) and nanomotors (maximizing the LCM of the
number of atoms in the circumferences of the armature and armature
sleeve for smoothest operation).

You'll find some practical applications listed here, although they're
computer-oriented rather than physical applications.
http://www.utm.edu/research/primes/notes/1257787.html

Hi, gw-ga:

This biological application of prime numbers is speculative -- it has
not been proven from the fossil record -- but appealing.  It is
suggested that the 13- and 17-year periodic emergence of cicadas (for
mating) is due to coevolution with predators (that lost the
competition and became extinct).

[Cicadas appear in their prime]
http://www.nature.com/nsu/010726/010726-3.html

The significance of 13 and 17 being primes is that a predator cannot
have any "divisor" lifecycle length except 1 or the same as the
cicadas.  If the cicadas lifecycle length were composite, then a
predator could possess a nontrivial divisor lifecycle length and still
manage to emerge synchronously with its prey.

This idea has been around for a long time, but the Nature article
notes some recent mathematical simulation using a predator-prey model
to support the theory.

regards, mathtalk-ga

Since Physical should involve physics here is a link the explains some
of the connections at a fundamental atomic level with that of prime
numbers (actually riemann function) in a weird way :

http://www.maths.ex.ac.uk/~mwatkins/zeta/surprising.htm

Other than I am not sure what role prime number play in music.Here is
a link that talks about aesthetics of primes:

http://www.2357.a-tu.net/