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.
QUESTION: Was Ljunggren’s equation solved by Lagrange and/or Lebesgue and/or Lucas?