Solvability of 2kk = 2aa x+2bb

Suppose k, x, and b are positive integers, and a is a nonnegative integer such that k=a+b. In this paper, we will prove 2kk = 2aa x+2bb if and only if x=a=1. We do this by looking at different cases depending on the values of x and k. We use varying techniques to prove the cases, such as direct proof, verification through Maple software, and a proof technique found in Moser's paper. Previous results from Hanson, Stanica, Shanta, Shorey and Nair are also used.