For a given odd integer I, it has been shown that the exponential Diophantine equation

has at most one solution, M and N, in integers, except when

5 or 13. In this paper, we show that many values of I are omitted, and derive necessary conditions on M and N for

to be assumed. In general, we find that when

is prime, then either M and N are relatively prime or M and N are bothsmall multiples of the same odd prime. We also study the divisibility properties of the assumed values of the equation. In particular, we show that if I is an assumed value of the equation and where q is prime and then M and N must have certain forms depending upon q and a.

exponential Diophantine equations, congruences, primitive roots.