The background of this question is this: Fermat proved that the equation, $$x^4+y^4 = z^2$$
has no solution in the positive integers. If we consider the near-miss, $$x^4+y^4-1 = z^2$$
then this has plenty (in fact, an infinity, as it can be solved by a Pell equation). But J. Cullen, by exhaustive search, found that the other near-miss, $$x^4+y^4+1 = z^2$$
has none with $0 < x,y < 10^6$.
Does the third equation really have none at all, or are the solutions just enormous?