Integer Pair and Prime Constant (Posted on 2009-10-24)

Determine all possible value(s) of a positive prime constantP such that the equation: 2^{X} – 3P = Y^{2} has precisely two distinct solutions in (X, Y), where each of X and Y is a positive integer.

From what I can tell so far, 5 is the only value of P for which there are exactly 2 solutions, which are (X,Y) = (4,1) and (6,7). I don't think any solutions are possible for P=2, and there may be only one solution for all remaining P.