Summary: 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 \((x,y,z)=(3,1,1),~(5,3,1),~(7,9,1),~(13,3,7)\). The conjecture implies a recent result of J. Lee [Acta Arith. 140, No. 1, 1–29 (2009; Zbl 1241.11124)] 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 \(\mathbb{Q}[\sqrt{x^2-2}]\) is 1, then \(x=3,5,7,13\).