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Wikipedia refers to the Diophantine equation $ x^2 + D = AB^n $ as an "equation of Ramanujan–Nagell type". It also says that "A result of Siegel implies that the number of solutions in each case is finite." My question is: What result of Siegel might this be referring to?

(Later in the text it has a similar statement about $x^2+D=Ay^n$, this time citing "results of Shorey and Tijdeman". I'm also curious about this, but mostly I want to know the Siegel result.)

Extended footnote: The reference provided (for both) unfortunately, presents difficulties. It is

Saradha, N.; Srinivasan, Anitha (2008). "Generalized Lebesgue–Ramanujan–Nagell equations". In Saradha, N. (ed.). Diophantine Equations. Narosa. pp. 207–223. ISBN 978-81-7319-898-4.

Sluething around, I found that this text is the proceedings for a conference held at the Tata Institute in 2005, presumably this one. No talk with the given name appears in the abstracts, but Anitha Srinivasan was a speaker (talk title: "On the equation $x^2+dy^2=bk^n$", which seems close enough). In any case, the fact that it is a conference proceeding does not give me much confidence that the citation will be adequate. But even if I did want to check, I don't know how to get my hands on the book without paying third parties. In fact, there does not even seem to be a way to buy the text from the publisher, whose website is now defunct.

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    $\begingroup$ They’re referring to his $1929$ finiteness theorem for integral points, anyway it follows from Mordell’s finiteness theorem for integral points on $y^2 = x^3 + k$ [aka Mordell curves] cause you just split into cases based on what $n$ could be mod $3$ and you find three equations $x^2 + D = A\cdot (B^m)^3$, $x^2 + D = AB\cdot (B^m)^3$, and $x^2 + D = AB^2\cdot (B^m)^3$. Now just multiply through by $A^2$, $(AB)^2$, and $(AB^2)^2$ in each case and you’re reduced to finding integral points on a Mordell curve (Mordell proved there are finitely many such, Baker gave a way to actually find them). $\endgroup$
    – alpoge
    Commented Feb 13, 2022 at 7:05
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    $\begingroup$ an English translation of Siegel's 1929 paper is here $\endgroup$
    – Carlo Beenakker
    Commented Feb 13, 2022 at 7:37
  • $\begingroup$ I've added the reference to Wikipedia: C. L. Siegel (1929). "Uber einige Anwendungen Diophantischer Approximationen". Abh. Preuss. Akad. Wiss. Phys. Math. Kl. 1: 41–69. $\endgroup$
    – Max Alekseyev
    Commented Feb 13, 2022 at 16:24
  • $\begingroup$ Thanks to everyone for the contributions :D $\endgroup$
    – Eric Nathan Stucky
    Commented Feb 13, 2022 at 17:02

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