Talk:Tetration

Recursive formula for nth super roots, not just the 3rd

I did a bit of messing around with the recursive formula for the 3rd super root and how it was derived, and I found that other super roots can be calculated with a similar formula, $x_{n+1}=\exp(W(W(x_{n}W({x_{n}}^{x_{n}}W(\cdots W(^{m-2}{x_{n}}\ln y))))))$, which gives the solution to $^{m}{x}=y$ when x₀ is set to 1 and as n approaches infinity. Although, it does get more complicated with higher values of m, and it only works for when m is a positive integer greater than 2. This is original research however, so this shouldn’t be included in the article. I’ll show how I derived it though. It goes something like this: ${\begin{aligned}^{m}x=y\rightarrow x^{^{m-1}x}=y\rightarrow e^{^{m-1}x\ln {x}}=y\rightarrow ^{m-1}x\ln {x}=\ln y\rightarrow x^{^{m-2}x}\ln {x}=\ln y\\\rightarrow e^{^{m-2}x\ln x}\ln {x}=\ln y\rightarrow e^{^{m-2}x\ln x}({^{m-2}x})\ln {x}=^{m-2}x\ln y\\\rightarrow ({^{m-2}x})\ln {x}=W(^{m-2}x\ln y)\cdots \end{aligned}}$Repeat the 4th to last steps shown here until you reach $^{1}x\ln(x)=x\ln(x)$ on the left side.

${\begin{aligned}x\ln x=W(xW(x^{x}\cdots W(^{m-2}x\ln(y))))\rightarrow \ln x=W(W(xW(x^{x}\cdots W(^{m-2}x\ln(y)))))\\\rightarrow x=\exp(W(W(xW(x^{x}\cdots W(^{m-2}x\ln(y)))))\end{aligned}}$You can then replace the x on the left side with x_n+1, and all of the x’s on the right side with x_n and set x₀ to 1. 107.9.36.50 (talk) 02:02, 23 June 2025 (UTC)

lower tetration

can you add lower tetration which is n copies of a combined by exponentiation, left-to-right. 46.98.144.28 (talk) 18:14, 1 October 2025 (UTC)

I think that you are talking about $$\operatorname {LowerTetration} (a,n)=a^{\left(a^{n-1}\right)}$$ for n > 0.

I don't see that naming this right-hand side to be "Lower tetration" adds much of anything to this article. Do you have sources that argue otherwise? —Quantling (talk | contribs) 19:40, 1 October 2025 (UTC)

i tried to write a reply with proof of existence but it got removed because it had a blacklisted link 46.98.144.28 (talk) 05:52, 2 October 2025 (UTC)

anyways, for example 2 lower tetrated to 4 is 256 46.98.144.28 (talk) 05:53, 2 October 2025 (UTC)

I agree that 2 "lower tetrated" to 4 is 256. Not coincidentally, it is the case that $2^{\left(2^{4-1}\right)}=256$ too. To include something like this definition or formula in the article, we need a reliable source — like a well-known textbook — that has a similar discussion. —Quantling (talk | contribs) 15:02, 2 October 2025 (UTC)

Although $2\leq \alpha <\omega \land 4\leq \beta <\omega \implies \alpha ^{\alpha ^{(-1+\beta )}}<(\alpha \uparrow \uparrow \beta )$, notice that $2\leq \alpha <\omega \land \omega <\beta \implies (\alpha \uparrow \uparrow \beta )=\omega <\alpha ^{\alpha ^{(-1+\beta )}}$. So lower=tetration beats tetration in the long run. JRSpriggs (talk) 21:33, 29 December 2025 (UTC)

I am not at all sure of this but, ... maybe the formula LowerTetration(α, β) = α^αβ−1 applies only for β < ω. Directly from their definitions, does lower tetration beat tetration in the long run? —Quantling (talk | contribs) 22:05, 29 December 2025 (UTC)

Letting α = 2. We see $(2^{2^{-1+\beta }})^{2}=2^{2^{-1+\beta }\cdot 2}=2^{2^{-1+\beta +1}}$ holds even when ω≤β.

Perhaps we could get the best of both worlds by redefining tetration as follows:

$\alpha \uparrow \uparrow 0=1$,

$\alpha \uparrow \uparrow (\beta +1)=\max(\alpha ^{(\alpha \uparrow \uparrow \beta )},{(\alpha \uparrow \uparrow \beta )}^{\alpha })$,

${\displaystyle \alpha \uparrow \uparrow \lambda =\sup(\{\alpha \uparrow \uparrow \beta$ :\beta <\lambda \})} when λ is a limit.

OK? JRSpriggs (talk) 23:31, 29 December 2025 (UTC)

If there's a reasonable source that discusses mixing tetration and lower tetration in that way, we could discuss it in the article. Because I'm always fearful of the hyperoperation cases where either argument is 0, I would define tetration and lower tetration for the other cases with

$$\alpha \uparrow \uparrow 1=\alpha$$

$$\alpha \uparrow \uparrow \beta =\sup _{0<\gamma <\beta }\alpha ^{\alpha \uparrow \uparrow \gamma }$$

$$\operatorname {LowerTetration} (\alpha ,1)=\alpha$$

$$\operatorname {LowerTetration} (\alpha ,\beta )=\sup _{0<\gamma <\beta }\operatorname {LowerTetration} (\alpha ,\gamma )^{\alpha }$$

And if it is noteworthy, we could add

$$\operatorname {BothTetration} (\alpha ,1)=\alpha$$

$$\operatorname {BothTetration} (\alpha ,\beta )=\max(\sup _{0<\gamma <\beta }\operatorname {BothTetration} (\alpha ,\gamma )^{\alpha },\sup _{0<\gamma <\beta }\alpha ^{\operatorname {BothTetration} (\alpha ,\gamma )})$$

(We could carefully add the cases with 0 too.)

—Quantling (talk | contribs) 15:50, 30 December 2025 (UTC)

I agree that LowerTetration needs to start with β = 1 because otherwise 1^α = 1 and then you are stuck there.

However, BothTetration works fine starting with β = 0 and α = 0.

If 2 ≤ α, then max (1^α, α¹) = max (1, α) = α.

Also max (1¹, 1¹) = max (1, 1) = 1. So for α = 1, BothTetration is the constant function 1 of β.

And max (1⁰, 0¹) = max (1, 0) = 1. So for α = 0, BothTetration is also the constant function 1 of β.

Notice that BothTetration acts like LowerTetration when α = 0 or β = the successor of a limit ordinal, and it acts like Tetration otherwise.

When α = 0, ordinary Tetration gives 1 when β is even and 0 when β is odd; so it cannot be continuous.

OK? JRSpriggs (talk) 15:51, 31 December 2025 (UTC)

real and complex heights

Has anyone defined the value of tetration for a height z that is a complex number, perhaps restricted to those with a real part in [0, 1), as follows? For some large integer n compute ⁿa. Use z to adjust ⁿa to get a value that approximates ^(n+z)a. Iteratively apply log_a to this value n times, to get an approximation for ^za. Finally take the limit using this process as n → +∞.

For example, we could use a z-based adjustment that makes sense for the case that the limit ^∞x = y exists. In particular, starting with x^y = y, we can write

x^{y + ε} = x^y (x^ε) = y (1 + ε ln x + O(ε²)) = y + εy ln x+ O(ε²) = y + ε ln y + O(ε²).

Iterating, by taking x raised to that and repeating z times in total, gives a final value of y + ε (ln y)^z + O(ε²).

The big win is that (ln y)^z is easy to interpolate for non-integer values of z. We can use ε = ⁿx − y and then define

${}^{z}x=\lim _{n\to +\infty }\log _{x}^{[n]}\left(y+\left({}^{n}x-y\right)\left(\ln y\right)^{z}\right)$

where log_x^[n] means apply the logarithm (base x) for n times, and where y = ^∞x. We hope that the limit makes sure that the O(ε²) terms are negligible.

So, for the case that y exists, does this work for non-integer values of z, is it analytic, does it somehow extend to values of x for which the limit ^∞x does not exist, and/or is there a textbook that thinks any of this is notable? —Quantling (talk | contribs) 18:01, 17 November 2025 (UTC)

Undue weight in section Complex heights

This section appears to give undue weight to the work of Vincent Vey. A neutral reformulation and possible reduction would be welcome. Furthermore, the only reference currently provided by Vey appears to be a self-published PDF that reads more like a set of personal notes than a scientific article.Lonico978 (talk) 22:21, 27 December 2025 (UTC)

Removed, since it is not peer-reviewed. –LaundryPizza03 (dc̄) 08:01, 19 March 2026 (UTC)

Index order

The index order for iterated and infinite exponentials seems backwards, especially for the latter. Since when computing, one starts at the top of the tower and work downwards, (as currently presented) it means starting from the last index(!). How is that even handeled in the infinite case?? ~2026-27517-79 (talk) 12:59, 7 May 2026 (UTC)

It is handled as a limit. Each finite-height tower can be evaluated (starting at its top and progressing to its bottom) to some value. Repeat for every possible finite height. If the limit as the height grows unbounded of that sequence of values converges to a limiting value then that limiting value is said to be the value of the infinite-height tower. —Quantling (talk | contribs) 21:37, 7 May 2026 (UTC)

I meant notation-wise, since one would have to start from (a_{inf-1})^(a_inf) and descend down, but AFAIK inf-1=inf so what is one to make of that?

I can't think of any other iterated operation on subscripted elements where one starts from the highest indexed one.

My point is that the tower top element should have index 1, the next beneath it 2, and so on (ie. …a3^a2^a1, showing that the ellipsis must be on the low end for an infinite tower). ~2026-27517-79 (talk) 23:14, 7 May 2026 (UTC)

I think I now see what you are saying. Yes, I agree it would be neater if we identified the values in the tower top down with increasing integers as indices. And, yes, that means the ellipsis that means "and so on" would be at the bottom. We'd use something like $\color {green}{\cdot ^{\cdot ^{\cdot ^{a^{a}}}}}$ or $\color {green}{\cdot ^{\cdot ^{\cdot ^{a^{a^{a}}}}}}$ rather than expressions like $\color {red}{a^{a^{\cdot ^{\cdot ^{\cdot }}}}}$ and $\color {red}{a^{a^{a^{\cdot ^{\cdot ^{\cdot }}}}}}$ that have the ellipsis at the top of the tower. And I think we'd also avoid having an a after the ellipsis, as in $\color {red}{a^{\cdot ^{\cdot ^{\cdot ^{a^{a}}}}}}$.

That all seems reasonable to me ... but you and I don't count — we'd have to find a reliable textbook or two that backs us up. —Quantling (talk | contribs) 14:47, 8 May 2026 (UTC)

Good that you see my point.

I'm no mathematician so I wouldn't know which text books that treat tetration (my books from engineering grad school certainly did not), but I would be amazed if anyone actually working in the relevant field(s) would index from bottom up (as in the article) as it's so obviously just the wrong way. IOW: finding a source should be easy for those who know where to look.

What would surprise me more is if the current notation is backed by textbook references, unless it's mentioned as a curiosity and the author don't really have expertise on the subject, and the indexing order came about more by coinidence (eg. due to the order of typing in LaTeX is bottom-up). Are there actually any credible sources that use the current index ordering? ~2026-27517-79 (talk) 20:08, 10 May 2026 (UTC)

Like you, I have no textbooks that discuss tetration, especially infinite tetration, and thus no notation that I can draw from them. We could simply wait until an editor who does joins this conversation.

Or, I suppose, we could boldly remove the existing notation that we find troubling, on the grounds that we don't have evidence that it is used in textbooks. Though we don't have evidence of "our" notation either, so I wouldn't add it. Maybe if you do the deletion carefully, we can still convey the concepts that the article is trying to convey. Would you like to try? You could boldly edit the article, or make proposals here on the talk page. —Quantling (talk | contribs) 21:32, 10 May 2026 (UTC)

What I had in mind is not remove anything, just edit the indexes so they start from 1 on the top element/exponent and increase downwards; of course, in case of the infinite tower, the ellipsis must be moved as well. I think the only place where indexed exponents occur is in the table in the Terminology section.

I don't see how any concepts the article conveys could be "distorted" by such a change, it's "just notation" (but it still ought to be correct, ie. indexes corresponding to association/evaluation rules).

Based on your previous comment I think you are more capable than me regarding the actual editing, so I nominate you for the task :) ~2026-27517-79 (talk) 21:51, 11 May 2026 (UTC)

I didn't see a way to modify the article to not use ellipses... we're going to have to use them at the right (top) or at the left (bottom) of those towers. I opted to have them at the left. Let's see what other editors think. —Quantling (talk | contribs) 22:25, 11 May 2026 (UTC)

Good job! It looks exactly like I had in mind. I didn't mean to imply somehow that ellipses wouldn't be necessary, they are the std. notation for "continue this pattern"; the point was just to start indexing from the top (right) and then the ellipsis is most naturally placed at the low left (as you did) following common notation for eg. progressions and series ~2026-27517-79 (talk) 10:46, 13 May 2026 (UTC)