We prove that if Cay(G;S) is a connected Cayley graph with n vertices, and
the prime factorization of n is very small, then Cay(G;S) has a hamiltonian
cycle. More precisely, if p, q, and r are distinct primes, then n can be of the
form kp with k < 32 and k not equal to 24, or of the form kpq with k < 6, or of
the form pqr, or of the form kp^2 with k < 5, or of the form kp^3 with k < 3.
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