Suppose f is a computable function from a recursively enumerable set U to the natural numbers and that L,K are r.e. subsets of U. Is f(L-K) a difference of r.e. subsets? The motivation comes from

Primes occurring as orders of elements of a finitely presented group

A positive answer would mean that the theorem proposed in HW's nice answer is 100% correct. Otherwise the $\epsilon$-clarification in my answer is actually needed.