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Pretty much what the title says. To be more general, try to solve $x! = n$. I have tried for many hours and only ended up with a headache, is there any good/decent/practical way of solving such an eqation? I could not find anything about this on the internet.

Tl;dr What would be an exact solution for x, when x! = n?

Zaizer zazza
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2 Answers2

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The Gamma function generalizes factorials. For this particular numerical question you can ask Wolfram Alpha to $$ \text{ solve } x! = 10 $$

It tells you $$ x ≈ 3.39008 $$ which makes sense: it's between $3$ and $4$.

https://www.wolframalpha.com/input/?i=solve+x!+%3D+10

Ethan Bolker
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Do you know about Gamma function that extends notion of factorial to all real and complex numbers? In general case the answer can be given only via it.

Hasek
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    This doesn't tell you how to find the inverse though. – Simply Beautiful Art Feb 20 '17 at 21:20
  • Even if it's hard to tell precise in analytic terms about its inverse for any particular argument you may try to find it numerically, check this MathOverflow question for more details: https://mathoverflow.net/questions/12828/inverse-gamma-function – Hasek Feb 20 '17 at 21:27
  • This is definitely what Ethan Bolker done in his answer -- use CAS for solving numerically equation with Gamma function. I believe this is the only possible way. – Hasek Feb 20 '17 at 21:30
  • Yes, but you haven't explained any of that! – Simply Beautiful Art Feb 21 '17 at 01:47
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    Your post would be better provided as a Comment, as you provide less of an answer than a proposed limitation on how it can be answered (similar to comments offered about the same time). – hardmath Feb 21 '17 at 01:54