Pseudopowers and primality proving

The so-called pseudosquares can be employed in very powerful machinery for the primality testing of integers N . In fact, assuming reasonable heuristics (which have been confirmed for numbers to 2) they can be used to provide a deterministic primality test in time O(log N), which some believe to be best possible. In the 1980s D.H. Lehmer posed a question tantamount to whether this could be extended to pseudo r powers. Very recently this was accomplished for r = 3, which naturally leads to the question of whether anything can be achieved for r > 3. In this paper we show how these earlier results can be extended to all prime values of r.