Questions tagged [power-towers]

For questions pertaining to power towers: expressions like a^(b^(c^d))), which result from iterated exponentiation. The "hyperoperation" tag may be appropriate, too.

Power towers are obtained by iterated exponentiation; an archetypal example is:

$\large a^{b^{c^d}}$

Power towers have been studied a lot; there is particularly many information on:

  • Modular arithmetic with power towers;
  • Convergence of "infinite" power towers; for example: $$\large\sqrt2^{\sqrt2^{\sqrt2^{\cdots}}} = 2$$

If we have a (finite) power tower with the same number repeated, such as the one with $\sqrt 2$ above, we speak of tetration. Tetration is the fourth hyperoperation, so if applicable, also include the tag.

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Solve X=sqrt(A)^sqrt(A)^sqrt(A)^..............infinty?

If $X= \newcommand{\W}{\operatorname{W}}\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{\sqrt{A}^{.{^{.^{\dots}}}}}}}}}}} $ then what is the value of $X^2-e^{1/X}$ ?
Rohit
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Difference between calculator and google calc for power

I tried to compute the power of 2^2^2^2 on google calculator and my casio calculator but both are giving different results. same is true for 3^3^3. Please explain me the difference between two expressions.
shiv garg
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Compute modulo of prime power tower

I have a prime number $p$, and I need to compute $p$ ↑↑ $k$ mod $m$. here p ↑↑ k can be written as p ^ (p ^ (p ^ (p ... k times))) for example - $p = 5, k = 3, m = 3$ 5 ↑↑ 3 = 5 ^ (5 ^ 5) % 3 = 2 Here $p$ is prime number, $2 <= p <= 10 ^ 8$, and…
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How to efficiently find a prime number $x$ raised to the power $x$ $k$ times modulo $m$?

How to efficiently find a prime number $x$ raised to the power $x$ $k$ times modulo $m$? In other words, how to find $ \underbrace{x^{x^{...^{x}}}}_k \mod m$, where $x$ is prime?
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Confused with exponent rules?

What power would I need to raise $4^{2^{2^l}}$ to get $4^{2^{2^n}}$ where $n>l$? Very simple question but it has stumped me. I guess $2^{2^{n-l}}$ but I do not think that is right I am not sure?
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Why is $2^{2^{2^n}}$ not equal to $16^n$?

Why is $a^{b^{c^d}}$ not equal to ${(a^{b^c})}^d$ (for positive n)? For example, WolframAlpha seems to say that $2^{2^{2^n}}$ is not equal to $16^n$.
penalosa
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(Square root of 2) power (square root of 2) power...

The problem is to calculate A: A= sqrt(2)^sqrt(2)^sqrt(2)^... (Each one(not first and second!) is a power for the previous power) I used my usual(and only!) method: A=sqrt(2)^A It can't be correct because A can be both 2 and 4. What's wrong with…
Rima
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Differentiating $4^{x^{x^x}}$

$4^{x^{x^x}}$ Hi, I came across this question and would like to check whether I have it done correctly: $e^{x^3}\ln4=4^{x^3}(3\ln4\cdot x^2)$ is this the correct solution?
Kaka you
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Does there exist a prime that is a sum of two prime power towers?

Does there exist prime number of the form $$\huge 2^{3^{5^{\,.^{.^{.\,^{p_n}}}}}} + p_n^{p_{n-1}^{\,.^{.^{.\,^{3^{2}}}}}}$$ where $p_n$ is the $n$-th prime number(and both towers are running through the first $n$ primes in order), other than the…
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Inverse and named fixed values, with ↑↑?

The inverse of $+$ is $-$, of $\times$ is $/$ and of $\text{^}$ is Log. Continuing upwards hyperoperationally, what is the inverse of $↑↑$? Whats more somtimes values that are fixed are given names, some being labels such as: $2\times =…
alan2here
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Compute the product of digits of P

Give $P$ a integer number where $$P=2^{3^{4^{5^{\dots1000}}}}$$ Then Compute The product of dígits of $P$ Compute $P\pmod{5}$ for The segond i think its will be something…
cand
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if -a^(-b^-c) is a positive integer and a, b, and c are integers, then...

(a) a must be negative (b) b must be negative (c) c must be negative (d) b must be an even positive integer (e) none of the above
Murad
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Infinite Power Tower approximation: float error?

Desmos appears to plot it falsely using the $x^y = y$ definition, curving backwards. I've included a 50x exponent for comparison, which suggests no values flowing left in $x$-axis due to float error - but not so sure of the approximation method used…
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Is $i = e^{\frac{\pi}{2}e^{\frac{\pi}{2}^{.^{.^.}}}}$?

I came up with this and I am wondering if it is true, because it seems illogical that $i$ can be made from an infinite power tower of reals. The way I found this is the following: $$i=e^{\frac{\pi}{2}i}$$ From this rather useless definition of $i$…
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Correct my sketch of proof about the convexity of the "natural" power tower on $[1,\infty)$

Hi I want to show the following fact : Problem : Let $x\geq 1$ and $n\geq 1$ a natural number and define: $$f(x)={}^{2n}x=\underbrace{x^{x^{⋰^{x}}}}_{2n\text { times}}$$ Then we have : $$f''(x)\ge 0$$ My sketch of proof for $x\ge y\geq 1$ : We…
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