The equation $$x^2=2y^4-1$$ was studied and solved by Ljunggren, who showed that $(1,1)$ and $(293,13)$ are the only integer solutions. However, his proof was very difficult and L. J. Mordell thought there must be an easier proof. Nowadays, this equation can be rewritten in Weierstrass form and solved by an algorithm for elliptic curves. However, I was wondering what the original proof really was. Here are some of the references on Wikipedia that I can't seem to be able to open.
Draziotis, Konstantinos A. (2007), "The Ljunggren equation revisited", Colloquium Mathematicum 109 (1): 9–11, doi:10.4064/cm109-1-2, MR 2308822.
Ljunggren, Wilhelm (1942), "Zur Theorie der Gleichung x2 + 1 = Dy4", Avh. Norske Vid. Akad. Oslo. I. 1942 (5): 27, MR 0016375.
I have found a proof without the use of elliptic curves by Ray Steiner and Nikos Tzanakis, but the proof also requires some computational power, which Ljunggren didn't have access to. So if anyone knows a proof to this, it would be appreciated if you post it.