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I am looking for the following paper by Ljunggren, Wilhelm: "Zur Theorie der Gleichung $x^2 + 1 = Dy^4$", Avh. Norske Vid. Akad. Oslo. I., 1942 (5): 27

The main result of this paper which I am interested in is that the solutions in positive integers of the equation $x^2 + 1 = 2y^4$ are $(1,1), (239,13)$, so a reference summarizing Ljunggren's proof would be welcome as well.

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    $\begingroup$ This comment says "The original proof uses a very delicate variant of Skolem's p-adic method". $\endgroup$
    – Wojowu
    Commented Jun 11, 2020 at 18:29
  • $\begingroup$ Also user whose name resembles Jose Stgo. has a post on MathOverflow which summarizes a related paper and may have a link to your paper or near to your paper of interest. So search MathOverflow for It. Gerhard "You're In The Right Place". Paseman, 2020.06.11. $\endgroup$
    – Gerhard Paseman
    Commented Jun 11, 2020 at 18:35
  • $\begingroup$ Thanks, hopefully I'll find it. $\endgroup$
    – Random
    Commented Jun 11, 2020 at 19:32
  • $\begingroup$ An idea that has worked for me is to ask for a copy to someone who has recently cited the article in question. Many people (me included) keep scanned copies of articles which are impossible to find online. $\endgroup$
    – efs
    Commented Jun 11, 2020 at 23:21
  • $\begingroup$ Thanks, I'll try that! $\endgroup$
    – Random
    Commented Jun 12, 2020 at 14:04

1 Answer 1

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Two papers that simplify Ljunggren's proof:

An Elementary Proof for Ljunggren Equation (2017)

Simplifying the Solution of Ljunggren's Equation (1991)

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    $\begingroup$ I believe that the first paper has an error, specifically the claim at the bottom of page 2 that if (c,b,13k) is a Pythagorean triple then b/c is either 5/12 or 12/5. In any case, I am aware of other proofs but I am interested specifically in the orginal proof. I should probably have emphasized that in the original question. $\endgroup$
    – Random
    Commented Jun 11, 2020 at 19:29
  • $\begingroup$ $(16,63,65)$, $b/c=16/63$. $\endgroup$
    – Gerry Myerson
    Commented Jun 11, 2020 at 22:05

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