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Hölder's theorem says that Gamma function is very non-elementary, but it does not exclude the possibility that factorial is the restriction of some elementary function to natural numbers. The answer to this question on SE says that is impossible, but someone in the comment pointed out an issue, and I think the comment is right, which leads to the question:

  1. Is factorial the restriction of some elementary function to natural numbers?

Or maybe let's start with something easier:

  1. Does Stirling's formula $\displaystyle\sqrt{2\pi n}\frac{n^n}{e^n}$ ever produce an integer (asking for rational seems too difficult)? It certainly cannot produce integers twice.
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    $\begingroup$ If the answer to (2) were “yes” then $\pi$ and $e$ would be algebraically dependent, which nobody believes to be the case. The algebraic independence is unproved but would follow from Schanuel’s conjecture. $\endgroup$
    – KConrad
    Commented Feb 21, 2022 at 7:06
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    $\begingroup$ Presumably, $n=0$ doesn't count. $\endgroup$
    – Robert Israel
    Commented Feb 21, 2022 at 7:42
  • $\begingroup$ @KConrad I know. That's why I asked about integer not rational. Does that seem any easier? $\endgroup$
    – 183orbco3
    Commented Feb 21, 2022 at 18:56
  • $\begingroup$ @1830rbc03 no it's all hopeless at present. Do you agree the answer to (1) is far more likely to be "no" rather than "yes"? $\endgroup$
    – KConrad
    Commented Feb 21, 2022 at 19:06
  • $\begingroup$ @KConrad Of course I expect the answers to both to be "no". Is (1) open too? When I wrote the question I thought (2) must be easier than (1) but that may not be the case. Does (1) somehow follow from the non-elementarity of Gamma function? $\endgroup$
    – 183orbco3
    Commented Feb 21, 2022 at 19:18

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