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I would like to know about how to exactly do calculation with tetration, especially when the value $k$ is a negative value in: $a ↑↑ k$

I am aware of the process of tetration, which is repeated exponentiation. I would like to know the result when $k$ is negative, and how that is possible. Would also be helpful if I could also get an example with this.

Yours Sincerely, Aster17

Tsar Asterov XVII
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  • Tetration soon leads to extremely large numbers. I am not aware of any definition for other $k$ than positive integers that is consistent with the tetration defined in this case. If $a\uparrow \uparrow k$ is small enough it can be easily calculated recursively. – Peter Oct 24 '22 at 11:29
  • @Peter I know tetration quickly blows up. I was only wondering on how would tetration work when k is a negative integer, as I was curious about it, if something like that would even be possible. – Tsar Asterov XVII Oct 24 '22 at 11:30
  • Interestingly, research into canonical definitions for non-natural number heights of tetration is still ongoing. However, there are many resources which discuss this, including this Tetration Forum, and this "wiki" on how to calculate such (but be warned as the methods are rather in-depth). – Graviton Oct 24 '22 at 12:52
  • Related: How would tetration work for non integer numbers. – Graviton Oct 24 '22 at 12:58
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    @Graviton Thank you very much for the resources provided. These are going to be highly useful to me and I can't thank you enough. Am I allowed to post what I have found from reading these as an answer? – Tsar Asterov XVII Oct 24 '22 at 16:03
  • @Aster17 Happy to assist. I've asked many-a questions on tetration here before, so I know the curiosity. So, be my guest, if you feel like you have found the answers you were looking for! It's your question after-all. – Graviton Oct 25 '22 at 11:46
  • @Graviton thanks for the assisstance, it's extremely helpful for my little own theory thing I'm writing. I'll post the answer here to finish the question. Thanks, Aster17 – Tsar Asterov XVII Oct 25 '22 at 14:46

1 Answers1

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So,

It seems tetrating values to a negative value seems to be a still studied phenomenon. But, the answer seems to be the following:

$a ↑↑ -1$
$=> log$ a $ (a^0) $
$=> log$ a $ (1) $
$=> 0 $

It seems to remain undefined for any negative integer aside from -1, due to the fact that there is no finite number which you can raise a number to that will lead to 0.

So, the answer is undefined for now, until tetration itself gets... you could say, well-defined!

Bad puns aside, I would like to thank Graviton for leading me to the right resources for getting the right answer and I will link them down below for further information.

Cheers,
Aster17

References:
https://math.eretrandre.org/tetrationforum/index.php
https://en.wikipedia.org/wiki/User:MathFacts/Tetration_Summary
How would tetration work for non integer numbers.

Tsar Asterov XVII
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