the class numbers of the real cyclotomic fields ${\bf q}(\cos(2\pi/{p^n}))$ are very difficult to compute. indeed, they are not known for a single prime $p>67$. we analyze these class numbers using the cohen-lenstra heuristics on class groups and are led to make the following conjecture: for all but finitely many primes $p$, the class number of ${\bf q}(\cos(2\pi/ {p^n}))$ is equal to the class number of ${\bf q}(\cos(2\pi/ {p}))$ for all positive integers $n$. it is possible that there are no exceptional primes $p$ at all. work in progress to test this conjecture empirically will also be discussed. this is joint work with joe buhler and carl pomerance.