It is shown that there are infinitely many natural numbers which differ from the next four greater perfect squares by a perfect square. This follows from the determination of certain families of solutions to the diophantine equation 2(b2 + 1) = a2 + c2. However, it is essentially known that any natural number with this property cannot be 1 less than a perfect square. The question whether there exists a number differing from the next five greater squares by squares is open.