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, St{a}nic{a}, Shanta, Shorey and Nair are also used.