Did Lagrange and/or Lebesgue and/or Lucas solve Ljunggren's equation?

Solution 1:

Two remarks before I turn to the answer: Ribenboim's references are notoriously unreliable, Dickson's references are almost always accurate.

In the case at hand, Dickson's (2) is the equation $2x^4 - y^4 = z^2$. Its integral solutions correspond to rational solutions of $2X^4 - 1 = Z^2$. The determination of rational points on curves of genus $1$ (in this case) is not very hard.

Determining the integral solutions of $2X^4 - 1 = Z^2$ is difficult; I am not sure what Dickson wanted to say, but clearly he does not claim that anyone had proved that the only integral solutions of your equation are $(1,1)$ and $(239,13)$.