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?

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    $\begingroup$ just gonna point out the L-tendency to the names $\endgroup$ – Jacob Claassen Sep 4 '17 at 20:09
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    $\begingroup$ not enough for an L-series $\endgroup$ – Lubin Sep 4 '17 at 21:55
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    $\begingroup$ @JacobClaassen: Kinda reminds me of the same with Superman's Lois Lane, Lana Lang, and Lex Luther. What the heck with the L's. $\endgroup$ – Tito Piezas III Sep 6 '17 at 16:09
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    $\begingroup$ @TitoPiezasIII: Speaking as a composer and music director, the L's make perfect sense for the femme(s) fatale(s), because of the liquidness and lusciousness of the sound ("L" is a voiced consonant). For Lex…? I got nothing. $\endgroup$ – Kieren MacMillan Sep 6 '17 at 17:08
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    $\begingroup$ @KierenMacMillan: The things I learn in MSE. :) $\endgroup$ – Tito Piezas III Sep 7 '17 at 2:27

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