I. It's widely believed that there's no nice function f : Z+ —> Z+ whose values are all prime: f(Z+) ⊂ Primes , where Primes = {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97,...}. Are there any actual theorems that say as much — necessarily defining "nice" ? Or at least restricting the classes of nice functions for which this might be possible? How about heuristics indicating why this should be hard? * * * II. What kinds of nice functions f : Z+ —> Z+ are known to have infinitely many primes in their image: |f(Z+) ∩ Primes| = oo ??? Dirichlet's famous 1837 theorem states that every arithmetic sequence of form {X_n = A + B n}, where A, B are relatively prime integers, contains infinitely many primes.* But I don't know of any other cases. Is that because they aren't known? —Dan ————— * I once read the proof (as a very readable chapter in Elliptic Curves by Anthony Knapp) and thought it astonishing innovative, but especially for that long ago!