Tetration Forum - All Forums

My introduction and Some Approximations of Kneser Tetration on Desmos I made

Tue, 07 Jul 2026 13:22:44 +0000

Hello! My name is Raees. I have been researching tetration and hyperoperations for 5 months. I am currently studying in High School. My birthday is July 2nd. I was born in India. My Hobby is currently researching Tetration and Hyperoperations(since there is so much to explore!) and making numerical models of functions in Desmos. I am mostly self-taught in math.

Anyways, I have tried to make some implementations of Kneser Tetration for Complex heights and a fixed base on Desmos(specifically the function sexp(a+bi), with the data computed using PARI/GP and the program fatou.gp made by sheldonism):

https://www.desmos.com/calculator/hsjtrcgqgs (Test Version to check my idea, works for imag(x) ∈ [-5,5], base e)

These are the Current Versions for a variety of Bases(which technically works for all complex numbers, but i reccommend the real part staying low ):
https://www.desmos.com/calculator/n1prudguef (Current version, base 1.5)
https://www.desmos.com/calculator/anektsk1pb (Current version, base 2)
https://www.desmos.com/calculator/d4frqmds2e (Current version, base 2.5)
https://www.desmos.com/calculator/otrjcsnhoe (Current version, base e)
https://www.desmos.com/calculator/rwfsrv1lte (Current version, base 3)
https://www.desmos.com/calculator/dtvyzocshm (Current version, base Pi)
https://www.desmos.com/calculator/8jbdph3pgv (Current version, base i(I'm not actually sure if its Kneser, i got the data from fatou.gp made by sheldonism))

I also made another approximation for real heights and bases ∈ [1.5,3.5]: https://www.desmos.com/calculator/12qaelj9vz(data computed from fatou.gp)

_________________________________________________________________________________________________________________________________________________

My current goal is to make Kneser Tetration for a lot of bases in Desmos. I discovered a path were I think i could achieve an approximation of Kneser tetration(but I need to research and refine my method, also a little sneak peak into the method: https://www.desmos.com/calculator/wonk57usxr).
Feel free to ask me any questions about my approximations of Kneser Tetration on Desmos!(also I am known as DubiumTetration on Reddit, and i uploaded one of my graphs on reddit:"https://www.reddit.com/r/desmos/comments/1tzgjnl/kneser_tetration_for_base_e_for_complex_heights/")

test

Wed, 24 Jun 2026 17:38:14 +0000

can normal tetration be defined as a limit of product tetration?

Wed, 27 May 2026 17:00:54 +0000

it is hypothesized that if you take infinitely many arbitrary base logarithms of product tetration with the same base, you will eventually get normal tetration with the base you chose.
but is it really true? someone please let me know.
here's an image to make things clearer:

image of tetration.png (Size: 85.42 KB / Downloads: 163)

Recurrence relations and differential equations

Wed, 06 May 2026 06:58:49 +0000

I'd like to hear your thoughts on why recurrence relations are considered the discrete version of differential equations.
I'm trying to generalize recurrence relations and autonomous differential equations.
I think that \(\mathbb{R}_+\)-actions are the straightforward continuous version of \(\mathbb{N}\)-actions, so I checked whether an orbit of a \(\mathbb{R}_+\)-action is always a solution (an integral curve) of an autonomous differential equation, and the opposite.

\(\gamma: \mathbb{R} \to \mathbb{R}^n\) is an orbit of a \(\mathbb{R}\)-action if and only if \(\gamma(s) = \gamma(t) \implies \gamma(s+u) = \gamma(t+u)\) for all \(u \in \mathbb{R}\).
\(\gamma\) is an orbit of a \(\mathbb{R}_+\)-action if and only if \(\gamma(s) = \gamma(t) \implies \gamma(s+u) = \gamma(t+u)\) for all \(u > 0\).
\(\gamma\) is a solution of an autonomous differential equation \(\gamma'(t) = F(\gamma(t))\) if and only if \(\gamma(s) = \gamma(t) \implies \gamma'(s) = \gamma'(t)\). (Define \(F(x) := \gamma'(t)\) when \(x = \gamma(t)\). The \(F\) is well-defined on \(\gamma(\mathbb{R})\) because of the hypothesis.)

From the Picard–Lindelöf theorem, if \(F: \mathbb{R}^n \to \mathbb{R}^n\) is Lipschitz continuous, a solution \(\gamma: \mathbb{R} \to \mathbb{R}^n\) of a differential equation \(\gamma'(t) = F(\gamma(t))\) is always an orbit of a \(\mathbb{R}\)-action.
Trivially an orbit of a \(\mathbb{R}\)-action is an orbit of a \(\mathbb{R}_+\)-action.
If an orbit of a \(\mathbb{R}_+\)-action is differentiable, it's always a solution of an autonomous differential equation. (The proof is described below.)
Therefore let \(\gamma: \mathbb{R} \to \mathbb{R}^n\) is differentiable, then
\[
\begin{array}{ll}
& \text{there exists a Lipschitz continuous \(F: \mathbb{R}^n \to \mathbb{R}^n\) such that \(\gamma'(t) = F(\gamma(t))\)} \\
\implies & \text{\(\gamma\) is an orbit of a \(\mathbb{R}\)-action} \\
\implies & \text{\(\gamma\) is an orbit of a \(\mathbb{R}_+\)-action} \\
\implies & \text{there exists \(F: \mathbb{R}^n \to \mathbb{R}^n\) such that \(\gamma'(t) = F(\gamma(t))\)}.
\end{array}
\]
The opposites are not true.

An orbit of a \(\mathbb{R}\)-action that is not a solution of an autonomous differential equation of a Lipschitz continuous function:
Since
\[
\gamma(t) =
\begin{cases}
-t^2 & (t \leq 0) \\
t^2 & (t \geq 0)
\end{cases}
\]
is injective, the \(\gamma: \mathbb{R} \to \mathbb{R}^1\) is an orbit of \(\mathbb{R}\)-action.
Assume there exists a Lipschitz continuous \(F: \mathbb{R}^1 \to \mathbb{R}^1\) such that \(\gamma'(t) = F(\gamma(t))\).
Then there exists \(L \in \mathbb{R}\) such that \(|\gamma'(t_1) - \gamma'(t_2)| = |F(\gamma(t_1)) - F(\gamma(t_2))| \leq L|\gamma(t_1) - \gamma(t_2)|\) for all \(t_1, t_2 \in \mathbb{R}\).
Let \(t_1 \neq 0\) and \(t_2 = 0\).
We see \(\gamma'(t) = 2|t|\), so \(2|t_1| \leq L|t_1^2|\). Therefore
\[
\left|\frac{1}{t_1}\right| = \left|\frac{t_1}{t_1^2}\right| \leq \frac{L}{2}.
\]
\(\left|\frac{1}{t_1}\right| \longrightarrow \infty\) when \(t_1 \longrightarrow 0\), that is a contradiction.

A differentiable orbit of a \(\mathbb{R}_+\)-action that is not an orbit of a \(\mathbb{R}\)-action:
\[
\gamma: \mathbb{R} \to \mathbb{R}^1, \quad
\gamma(t) =
\begin{cases}
t^2 & (t \leq 0) \\
0 & (t \geq 0)
\end{cases}
\]
is an example of it.

A solution of an autonomous differential equation that is not an orbit of a \(\mathbb{R}_+\)-action:
\[
\gamma: \mathbb{R} \to \mathbb{R}^1, \quad
\gamma(t) =
\begin{cases}
0 & (t \leq 0) \\
t^2 & (t \geq 0)
\end{cases}
\]
is a solution of \(\gamma'(t) = F(\gamma(t))\) where
\[
F: \mathbb{R}^1 \to \mathbb{R}^1, \quad
F(x) =
\begin{cases}
0 & (x \leq 0) \\
2\sqrt{x} & (x \geq 0),
\end{cases}
\]
while \(\gamma(-1) = \gamma(0)\) and \(\gamma(-1+1) = 0 \neq 1 = \gamma(0+1)\).

Proof of that a differentiable orbit of a \(\mathbb{R}_+\)-action is a solution of an autonomous differential equation:
We'll prove the contraposition.
Let \(\gamma: \mathbb{R} \to \mathbb{R}^n\) is differentiable, \(\gamma(t) = (\gamma_1(t), \dots, \gamma_n(t))\). (\(\gamma'(t)\) denotes \((\gamma_1'(t), \dots, \gamma_n'(t))\).)
Assume \(\gamma\) is not a a solution of an autonomous differential equation.
Then there are \(t_1, t_2 \in \mathbb{R}\) such that \(\gamma(t_1) = \gamma(t_2)\) and \(\gamma'(t_1) \neq \gamma'(t_2)\).
Therefore \(\gamma'_i(t_1) \neq \gamma'_i(t_2)\) for some \(i\).
Define \(x = \gamma_i(t_1) = \gamma_i(t_2)\), \(\alpha = \gamma_i'(t_1)\), and \(\beta = \gamma_i'(t_2)\). We can assume \(\alpha < \beta\) w.l.o.g.
We know
\[
\forall \varepsilon_1 > 0, \exists \delta_1 > 0, \forall h \in (-\delta_1, \delta_1), \left|\frac{\gamma_i(t_1+h)-x}{h} - \alpha\right| < \varepsilon_1
\]
and
\[
\forall \varepsilon_2 > 0, \exists \delta_2 > 0, \forall h \in (-\delta_2, \delta_2), \left|\frac{\gamma_i(t_2+h)-x}{h} - \beta\right| < \varepsilon_2.
\]
Thus
\[
x + \alpha h - \varepsilon_1|h| < \gamma_i(t_1+h) < x + \alpha h + \varepsilon_1|h|,
\]
\[
x + \beta h - \varepsilon_2|h| < \gamma_i(t_2+h) < x + \beta h + \varepsilon_2|h|.
\]
Since \(\beta - \alpha > 0\) there are \(\varepsilon, \varepsilon' \in \mathbb{R}\) such that \(0 < \varepsilon < \varepsilon' < \beta - \alpha\).
Substitute \(\varepsilon_1 = \varepsilon\) and \(\varepsilon_2 = \beta - \alpha - \varepsilon'\), then for any \(0 < h < \min\{\delta_1, \delta_2\}\),
\begin{align*}
\gamma_i(t_1+h) &< x + \alpha h + \varepsilon_1|h| \\
&= x + \alpha h + \varepsilon h \\
&< x + \alpha h + \varepsilon'h \\
&= x + \beta h - (\beta - \alpha - \varepsilon')h \\
&= x + \beta h - \varepsilon_2|h| \\
&< \gamma_i(t_2+h).
\end{align*}
That is \(\gamma(t_1+h) \neq \gamma(t_2+h)\).
Therefore \(\gamma\) is not an orbit of a \(\mathbb{R}_+\)-action.

From these results, I lost sight of what is the relation between an autonomous differential equation and a \(\mathbb{R}_+\)-action (or a \(\mathbb{N}\)-action).
I was hoping you could help me find that relation.

the fraction iteration approximation of tetration

Tue, 14 Apr 2026 17:52:40 +0000

we approximate the arbitrary base logarithm with a rational function, namely (ax+b)/(cx+d).
we choose a to be 2, b to be -2, c to be ln(a1), d to also be ln(a1)
a1 is the arbitrary base of the logarithm.
when we're done, we use the iteration formula for (ax+b)/(cx+d).
and we get this graph: https://www.desmos.com/calculator/x6malmxjzt

Reviewing recent developments?

Wed, 31 Dec 2025 06:09:33 +0000

It’s been a long time since I participated in community discussions. Have there been any significant new developments regarding tetration lately? Also, has the community discussed the paper Holomorphic Extension of Tetration to Complex Bases and Heights via Schröder's Equation (https://doi.org/10.13140/RG.2.2.10348.48008)? What are its primary conclusions?
How this method compares to the Kneser?

extending normal tetration to real numbers using product tetration

Fri, 12 Dec 2025 18:49:13 +0000

we start by constructing the base e product tetration of x which is denoted as \(\text{prodtet}_{e}(x)\)
the base e product tetration has these recursive properties:
\(\text{prodtet}_{e}(x)=\text{prodtet}_{e}(x-1)*e^{\text{prodtet}_{e}(x-1)}\)
\(\text{prodtet}_{e}(x)=\text{W}(\text{prodtet}_{e}(x+1))\)
also, product tetration has this following graph: https://www.desmos.com/calculator/zkjiz8jwh0
if you're wondering how i extended product tetration to real numbers, i used lagrange interpolation on the very far negative side of product tetration and applied recursive formulas.
some example values of product tetration:
\(\text{prodtet}_{e}(-1)=\text{W}(1)=0.56714... \)
\(\text{prodtet}_{e}(0)=1\)
\(\text{prodtet}_{e}(1)=e\)
\(\text{prodtet}_{e}(2)=e^{e+1}\)
and one accepted non-integer value for product tetration:
\(\text{prodtet}_{e}(1/2)=1.5134...\)
and we can express product tetration to different bases by introducing the inverse function of product tetration which will be called the product super logarithm.
the product super logarithm will be denoted as \(\text{prodslog}_{e}(x)\)
the product super logarithm also has some noteworthy recursive properties of its own:
\(\text{prodslog}_{e}(x)=\text{prodslog}_{e}(xe^{x})-1\)
\(\text{prodslog}_{e}(x)=\text{prodslog}_{e}(\text{W}(x))+1\)
the product super logarithm also has a noteworthy graph (sorry if it's slow): https://www.desmos.com/calculator/sxjnv2212j
now for the base change formula of both the product tetration and the product super logarithm:
\( \text{prodtet}_{a}(x)=\frac{\text{prodtet}_{e}(\text{prodslog}_{e}(\text{ln}(a))+x)}{\text{ln}(a)}\)
\(\text{prodslog}_{a}(x)=\text{prodslog}_{e}(x*\text{ln}(a))-\text{prodslog}_{e}(\text{ln}(a))\)
here, i give an example of the base change formulas with base pi: https://www.desmos.com/calculator/kwsqnpe4ef
the proof of the base change formulas is left as an exercise to the reader.
now, back to the main topic of this post.
we can literally extend normal tetration to real numbers by using product tetration because product tetration grows faster than normal tetration.
here's how the process goes:
we extend both the product tetration and the product super logarithm to real numbers using any interpolation technique we consider good. (in this case, i considered the lagrange interpolation good.)
we take the product super logarithm of normal tetration and interpolate the result we get by doing so. (the interpolation still has to be good.)
we finally take the product tetration of the above interpolation to get normal tetration.
here's an example of this process being applied to base 2 tetration: https://www.desmos.com/calculator/r26svg3olm
the numerical accuracy on desmos is bad for product tetration and the product super logarithm.
and desmos can only calculate numbers up to \(2^{1024}\).
i hope there is a calculator that functions like desmos but with greater numerical accuracy and the ability to calculate greater numbers than \(2^{1024}\).

Confirmed: the constancy of the congruence speed holds in each squarefree radix

Mon, 01 Dec 2025 19:28:32 +0000

It has been a long time since my last post here, so I hope you will forgive the sudden appearance after one year of silence.
I would like to share a very compact one-page note that summarizes six fundamental equations describing the constant congruence speed of integer tetration bases (greater than 1 and not a multiple of the selected radix) in arbitrary numeral systems \(r > 1\).
These few equations condense the key results established in our referenced recent works, including the complete formulas for prime radices, the explicit radix-6 and radix-10 cases, and a couple of elegant identities valid for all squarefree \(r > 2\).

In particular, I am especially fond of identity (4). It highlights the structural role of the \(r\)-adic fixed point \(1_r\): it considers 5 different parameters on the left-hand side, including the height of the power tower, which only needs to be greater than 1, and for \(k = 0\) (by Mihăilescu's theorem) it returns only perfect powers of exact order \(c\) whose radix-\(r\) constant congruence speed is exactly \(t\). Furthermore, it includes hyper-1 as sum/difference, hyper-2 as product, hyper-3 as exponentiation, and finally the function \(V\) arises from tetration (i.e., hyper-4).

I am attaching these formulas as a figure; the one-page note is also available on Zenodo: Six fundamental equations for the constant congruence speed in any numeral system.

Comments, questions, and suggestions are of course welcome... it may also be interesting to discuss possible extensions or implications for related r-adic fixed-point structures or iterated-exponentiation phenomena (I have also developed a very rough key-exchange idea based on solving fundamental equations of the form \(y^t = y\) for sufficiently large odd values of \(t := t\)(\(r\)) in the ring of \(r\)-adic integers \(\mathbb{Z}_r\)).

Six fundamental equations (reduced).jpg (Size: 228.37 KB / Downloads: 336)

what we should call the hyperoperations in different languages

Sat, 15 Nov 2025 15:12:15 +0000

we already have the english names for hyperoperations: tetration, pentation, hexation, etc. but here are a few suggestions for what we should call the hyperoperations in different languages.
romanian: tetrație, pentație, hexație, etc.
icelandic: 4 rekstur, 5 rekstur, 6 rekstur, etc.
note: in icelandic, rekstur means operation.
german, french: same as english.
second note: we already have the russian, turkish, spanish, and italian names for hyperoperations.
mongolian: 4 үйл, 5 үйл, 6 үйл, etc.
third note: in mongolian, үйл means operation.
hungarian: 4 művelet, 5 művelet, 6 művelet, etc.
fourth note: in hungarian, művelet means operation.
you can add to the list if you know a language other than the ones mentioned here.

Tetration with complex bases

Thu, 13 Nov 2025 09:33:54 +0000

Seeing as how, for real bases, tetration is usually defined by computing the inverse Abel function (α^-1(x)), and mapping the region above the main line where Im(α^-1(x)) = 0 (also extended in the negative direction, where it would otherwise be complex) to the upper-half plane, I was wondering how this would be done for complex bases. Is complex-based tetration computed by just extrapolating from the coefficients of real tetration, or just by using the inverse Abel function, or a different method entirely?

Orbit-like maps on linearly ordered groups

Sat, 08 Nov 2025 07:47:16 +0000

I want to extend the indexes of \(a\) of a recurrence relation \(a_{n+1} = f(a_n)\) to arbitrary linearly ordered groups.
For a map \(a: \mathbb{Z} \to X, n \mapsto a_n\), there exists \(f: X \to X\) such that \(a_{n+1} = f(a_n)\) if and only if \(a_m = a_n \implies a_{m+k} = a_{n+k}\) for any positive integer \(k\).
When \(a: \mathbb{Z} \to X\) satisfies the above condition, the periods of \(a\), \(P(a) = \{m-n \mid a_m = a_n\}\) is a subgroup of \(\mathbb{Z}\).

To generalize to arbitrary linearly ordered groups, I will introduce some definitions.
For a map \(a\) defined on a linearly ordered group \(G\),
\(a\) is peep if \(a(g) = a(h) \implies a(fg) = a(fh)\) for all \(f > 1\).
\(a\) is bib if \(a(g) = a(h) \implies a(gf) = a(hf)\) for all \(f > 1\).
\(a\) is peep bib if \(a\) is peep and bib.
I define periods of \(a\) as
\begin{align}
P(a) &= \{g^{-1}h \mid a(g) = a(h)\} \\
&= \{p \in G \mid \exists g, h \in G, p = g^{-1}h\: \text{and}\: a(g) = a(h)\} \\
&= \{p \in G \mid \exists g \in G, a(g) = a(gp)\}.
\end{align}
A subgroup \(H\) of \(G\) is said to be fee if \(g^{-1}Hg \subset H\) for all \(g > 1\).

Let \(G\) be a bi-ordered group and \(a\) be a map defined on \(G\). (Edit: \(G\) need not to be bi-ordered, just need to be linearly ordered.) Then
\begin{equation}
\begin{array}{ccccc}
& & a\: \text{is}\: \textit{peep bib} & \implies & a\: \text{is}\: \textit{peep} \\
& & \Downarrow & & \Downarrow \\
P(a)\: \text{is normal} & \implies & P(a)\: \text{is}\: \textit{fee} & \implies & P(a)\: \text{is a subgroup} \\
\end{array}
\end{equation}
and the converses of these are not true.

For example, let \(F\) be the free group generated by \(a, b\) and bi-ordred in the way of Definition 3 of https://doi.org/10.4153/CJM-2003-034-2
Then the subgroup \(\langle a \rangle\) is not fee.
The fee subgroup \(\langle b^{F_+} \rangle\) is not normal (shown in the attached pdf
momo (4).pdf (Size: 223.12 KB / Downloads: 237) ).
We define a map \(a\) on \(\mathbb{Z}\) as
\begin{equation}
\dots, -7, -6, -5, 22, -3, 22, -1, 0, 1, 0, 1, 0, 1, 0, 1, \dots
\end{equation}
Then \(a\) is neither peep nor bib, and \(P(a) = 2\mathbb{Z}\).

Is my tetration-webspace still locked?

Mon, 20 Oct 2025 15:28:21 +0000

Dear friends -

it is about 6 weeks, that some correspondent from USA told me he cannot access the files on my website. It seemed so as well from Scotland - but the access from Germany seems to work (I can access everything by ftp:// and even by https:// ).
Now the admin of our small network https://helms-net.de tells me he had tried to reverse some changes he had made in spring, and I would like to check, whether international folks can now access my work again.

Could someone please try to access my math-homepage

        https://go.helms-net.de/math/tetdocs/index.htm

and if that is possible, then to download just a single *.pdf-file as test?

Please tell me the result here, or even better, mail me at

     mailto:gottfried.helms@t-online.de

(possibly include the error-message as well)

Thank you in advance -

Gottfried.

i have a seemingly reasonable definition for tetration

Sun, 19 Oct 2025 15:00:11 +0000

my definition of tetration goes like this:
1. i construct the xth super root of y by lagrange interpolation.
2. i evaluate my constructed super root function at a non-integer x value.
3. i finally construct my tetration function by using the y value i got for my super root function.
more details are given in this desmos graph: https://www.desmos.com/calculator/ooxd3v6twb
as can be seen in the desmos graph, i constructed the xth super root of 2 and i chose 3.5 as my x value.
that's how i constructed the 3.5th super root of 2 tetrated to x.
if you want to know the value for the 3.5th super root of 2 using my method and kneser's, here it is:
my method: \( \sqrt[3.5]{2}_{s}=1.45849906738\)
kneser's method: \( \sqrt[3.5]{2}_{s}=1.4584946676580992\)
my method and kneser's method give slightly different results. that might be because the lagrange interpolation from my method does not approximate the super root function as well as kneser's method. but maybe the approximation problem can be solved using limits.
here's what i mean: we take the limit of the xth super root as x goes to infinity.
we interpolate the function using the limit values and we apply recursive formulas to construct the xth super root.
it doesn't seem like it's easy to find recursive formulas for the xth super root.
like i said in the title of this thread, my tetration method SEEMS reasonable. but i don't yet know if it IS reasonable.
maybe someone finds a quick algorithm in desmos to compute super roots. the algorithm i found required a lot of iterations of newton's method. too many iterations of newton's method would be bad for my computer.
and another point i would like to make: it seems like i can continue on a similar path for pentation, hexation, etc.
and by continuing on a similar path, i mean adapting my tetration method for higher hyperoperations.

my proposed extension of the fast growing hierarchy to real numbers

Sun, 28 Sep 2025 18:15:55 +0000

i made this algorithm on desmos where i extended the fast growing hierarchy to real numbers: https://www.desmos.com/calculator/i9ue4cyvan
i took the superfunction of \(f_{2}(x)\) on the integer points and used lagrange interpolation on the left side of the superfunction.
and that's how i got \(f_{3}(x)\).
and you can go through the same process to get \(f_{4}(x)\), \(f_{5}(x)\), \(f_{6}(x)\), and so on...
but i don't even know how \(f_{\omega}(x)\) can be extended.
we first have to define the fast growing hierarchy for non-integer subscripts.
\(f_{n}(1)=2\) for all n. and we can choose to keep it true for non-integer values of n.
the only catch is that we don't yet have a full understanding of the fast growing hierarchy with real numbers.
we can't just start taking derivatives, plugging in values, or anything like that.
we have to actually understand the fast growing hierarchy.
if we understand the fast growing hierarchy well enough, we can even extend it to complex numbers.

product tetration

Wed, 24 Sep 2025 11:47:13 +0000

product tetration is the superfunction of \(f(x)=xa^{x}\).
it uses the lambert w function (also known as the product logarithm) to go in the negative direction.
product tetration requires three variables and is denoted as: \(\text{prodtet}(a,b,c)\)
product tetration can be interpolated through lagrange polynomials for extension to real numbers.
and it has this recursive property:
\(\text{prodtet}(a,b,c)= f(\text{prodtet}(a,b,c-1))\)
and this base case:
\(\text{prodtet}(a,b,0)= b\)
more details are given on this desmos graph: https://www.desmos.com/calculator/cnvdx5ce1r