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Recently, I've become interested in hyperoperations. I wondered what the equation y=x tetrated by x (x[4]x), which is the same as x pentated by 2 (x[5]2), would look like on a graph. To do this, though, would mean to tetrate a number by a non-integer value, and everywhere I've looked has just come up by saying that there just isn't any known way to compute this.

For a moment I tried to figure out a way but the properties of previous hyperoperations just don't carry over to tetration.

It's weird to me that this can be done with addition, multiplication, and exponentation but no further. Maybe we're just not used to looking at it in a certain way, I don't know, I'm not experienced in maths, I'm just curious.

Mainly, I just want to know if there is any/what research is being done on tetrating a number by a non-whole number/fraction. Or if there is any proposed way to calculate an answer to one of those problems.

Thanks in advance.

Joey Peluka
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  • Related: https://math.stackexchange.com/questions/238970/how-to-evaluate-fractional-tetrations and https://math.stackexchange.com/questions/56663/is-there-a-natural-way-to-extend-repeated-exponentiation-beyond-integers and https://math.stackexchange.com/questions/1638044/an-obvious-pattern-to-i-uparrow-uparrow-n-that-is-eluding-us-all - the real problem is that exponentiation is not associative – Henry Dec 16 '19 at 05:32
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    There seems to be a misunderstanding: it's not, that "there isn't any known way to compute this" - there are even a multitude of ansatzes, leading to nice-looking solutions. What is not known/decided is: which of the ansatzes are the most meaningful ones? Should my solution, which gives real numbers for real $x$ (in a certain interval) be taken, or should one take the solution which looks good too, but gives complex numbers as result? There are even program-codes openly available which compute such solutions for a wide range of arguments, so it's not a case "no one knows how to do" – Gottfried Helms Dec 16 '19 at 06:30
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    Perhaps you like to surf into the waves of the ocean at https://math.eretrandre.org/tetrationforum/index.php ... I say ocean because the plethora of posts is fairly un-organized but are inhabitated by colorful beasts underwater... – Gottfried Helms Dec 16 '19 at 06:33
  • Thanks for correcting me yes that's probably is a better way to put it. Which methods were you talking about? I know of linear/quadratic approximations but those are just approximations. What other methods exist? – Joey Peluka Dec 16 '19 at 17:53
  • @GottfriedHelms – Joey Peluka Dec 16 '19 at 19:21
  • For instance you might look for "Schroeder-function" or "Kneser-solution". I don't have an idea of your understanding level but I once tried to do an elementary introduction examining an ansatz using (formal) powerseries and their iterations. I'm an amateur too, so surely some professional could explain things better didactically and with sources, but so far you might look at http://go.helms-net.de/math/tetdocs/ContinuousfunctionalIteration.pdf – Gottfried Helms Dec 16 '19 at 20:08
  • ah, and to see a difference of methods you might look at http://go.helms-net.de/math/tetdocs/ComparisionOfInterpolations.pdf Also I've done a little bit in discussion linear vs polynomial interpolation here on MSE recently but don't have it at hand at the moment. I think there was a discussion in summer this year. – Gottfried Helms Dec 16 '19 at 20:11

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