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The Wikipedia page for Sophie Germain contains the following:

In the same 1807 letter, Sophie claimed that if $x^n + y^n$ is of the form $h^2 + nf^2$, then $x + y$ is also of that form. Gauss replied with a counterexample: $15^{11} + 8^{11}$ can be written as $h^2 + 11f^2$, but $15 + 8$ cannot.

WolframAlpha confirms that the equation $15^{11} + 8^{11} = h^2 + 11f^2$ has two solutions in the positive integers, $(h, f) = (935166, 841201)$ and $(h, f) = (1595826, 745391)$, and it is obvious that $15 + 8 = h^2 + 11f^2$ has no solutions in the integers.

However, how would one be able to confirm that $15^{11} + 8^{11} = h^2 + 11f^2$ has solutions in the positive integers without such a computational aid? In a related question, how might Gauss have come up with this counterexample in the first place?

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    $\begingroup$ Since Fermat's last theorem is all the rage these days, what with the Abel prize going to Andrew Wiles for his proof of the modularity conjecture, I have been to a few lectures on the subject the last few days, most recently today, by Wiles himself and others. And if there's anything I've learned about the mathematicians of the past from these lectures, it's that they were really practiced at calculating stuff like this. That being said, if Gauss noticed something that pointed him towards this counterexample, it would be cool to know what that might have been. $\endgroup$
    – Arthur
    May 25, 2016 at 18:29
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    $\begingroup$ jstor.org/stable/2324363?seq=1#page_scan_tab_contents shows the beginning of the article by Waterhouse. I should have access online but it is being stubborn so far. $\endgroup$
    – Will Jagy
    May 25, 2016 at 18:38
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    $\begingroup$ maa.org/sites/default/files/pdf/upload_library/22/Ford/… $\endgroup$
    – Will Jagy
    May 25, 2016 at 18:41
  • $\begingroup$ I'm sure that by his work, Gauss hat a lot of "gut feeling" knowledge about quadratic residues, and that may have helped $\endgroup$
    – Hagen von Eitzen
    May 25, 2016 at 20:00
  • $\begingroup$ And don't forget that Gauss was some kind of a prodigy calculator. Just think that he calculated with his own hands - what else ? - the orbit of the asteroid Xeres ! $\endgroup$
    – nguyen quang do
    May 26, 2016 at 6:24

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