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Wolfram Mathworld lists several transcendental numbers such as $$\Gamma{\left(\frac{1}{3}\right)},\Gamma{\left(\frac{1}{4}\right)},\Gamma{\left(\frac{1}{6}\right)}$$ I don't see the reason why Wolfram Mathworld skips $\Gamma{\left(\frac{1}{5}\right)}$.

Is it because it has not yet been proven that $\Gamma{\left(\frac{1}{5}\right)}$ is transcendental? If so, what makes it hard to prove the transcendentality of $\Gamma{\left(\frac{1}{5}\right)}$?

Larry
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    I expect it is unknown. this suggests that transcendence is known for very few values. – lulu Dec 19 '18 at 01:23
  • So we can't approximate it using quadratically convergent arithmetic-geometric mean. Is it the reason why it is difficult? – Larry Dec 19 '18 at 01:31
  • I suspect the answer to this is very technical - it doesn't look like it was straightforward to show that the numbers you list are transcendental, and it could either be that the methods used were just inherently focused on one value or that there's some subtle problem in the techniques that prevents them from generalizing at all. In either case, it'd likely be necessary to understand why those numbers are transcendental in order to understand why it's hard to show that for $\Gamma(1/5)$. – Milo Brandt Dec 19 '18 at 01:31
  • @Brandt: Helpful links or general explanation would be fine. I am not necessarily looking for a technical answer (it would be great if someone provides a detailed explanation). – Larry Dec 19 '18 at 01:39
  • The question really should be: what makes it easy to prove that the others are transcendental? In general proving that numbers are transcendental is hard. – Matt Samuel Dec 19 '18 at 02:13
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    Founds this - https://math.stackexchange.com/questions/2360504/how-justify-that-gamma-left-frac16-right-is-a-transcendental-number?rq=1 –  Jan 02 '19 at 04:43

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