ON THE DIOPHANTINE EQUATION 2(x) = x(2)

In this paper, we show that the only positive integer solutions of the equation 2(x) = x(2) + y(2) - 2 are ( x, y) = ( 3, 1), ( 5, 3), ( 7, 9). We propose also the following conjecture: the equation 2(x) =y(2) + z(2) (x(2) - 2) , where y, z are odd positive integers and x is a positive integer such that x(2) - 2 is a prime number, has the only solutions ( x, y, z) = ( 3, 1, 1), ( 5, 3, 1), ( 7, 9, 1), ( 1 3, 3, 7). The conjecture implies a recent result of Lee [ 4] which states that if x 2 - 2 is an odd prime number such that the class number h(x(2) - 2) of the quadratic field Q[`root x(2) - 2] is 1, then x = 3, 5, 7, 13.