Ljunggren’s Diophantine equation is $$X^2+1=2Y^4.$$ In 1942, he solved it using extremely difficult means; Mordell asked for a simpler proof, so people have been trying (unsuccessfully) ever since.
But in My Numbers, My Friends, Ribenboim says that the equation ”had been solved by Lagrange (1777) by the method of descent.” Additionally, Dickson (History, Vol. II) writes “A. Cunningham noted that the solution of (2) by Lebesgue and Lucas appears to be complete and to indicate that the only integral solutions of $x^2-2y^4=-1$ are $(1,1)$ and $(239,13)$”. I’ve been told by several mathematicians (including Noam Elkies, in an email of just over a week ago), that there is no elementary solution.
So…
QUESTION: Was Ljunggren’s equation solved by Lagrange and/or Lebesgue and/or Lucas?
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 $(239,13)$.