# Tagged Questions

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### Appear the digits in the number $3^{3^{3^3}}$ approximately equally often?

Does the number $$3\uparrow \uparrow 4 = 3^{3^{3^3}}$$ contain the digits 0-9 approximately equally often ? With "approximately" I mean that a chi-squared test for equal distribution would produce ...
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### Prime factors of $2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}$

Are there any useful restrictions to the prime factors of the number $$2^{2^2}+3^{3^3}+5^{5^5}+7^{7^7}$$ The two smallest are 6771419 and 72153167 , which I found by trial division. The number is ...
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### Comparison between two tetrations

For a given natural number n, what ist the least number m, such that $$e \uparrow \uparrow m > \pi \uparrow \uparrow n$$ It seems that m = n + 1 is the desired number. Is this true for all n ?
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### Two more numbers involving tetration with unknown prime factors

I have two more numbers for which I search prime factors : $$6 \uparrow \uparrow 4 + 7 \uparrow \uparrow 4$$ and $$7 \uparrow \uparrow 4 + 10 \uparrow \uparrow 4$$ I searched upto $10^9$ and did ...
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### Is there a proof that the number $2 \uparrow \uparrow m+3 \uparrow \uparrow n$ is always squarefree?

I searched prime factors of the numbers $$z(m,n) := 2 \uparrow \uparrow m + 3 \uparrow \uparrow n$$ where $m,n\ge1$ Interestingly, z(3,3) is prime, the largest prime I found so far and probably the ...
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### Does $a \uparrow \uparrow (n+1)-a \uparrow \uparrow n$ divide $a \uparrow \uparrow(n+2) - a \uparrow \uparrow(n+1 )$?

Does $$a\uparrow \uparrow (n+1) - a\uparrow \uparrow n$$ divide $$a\uparrow \uparrow (n+2) - a\uparrow \uparrow (n+1)$$ for all $a,n \ge2$ ? The case n = 0 is easy : $$a^a-a=a(a^{a-1}-1)$$ and ...
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### Stationary prime factor 34276387 , confirmation and generalization wanted.

Sheldon L found out, that $$34276387$$ is a prime factor of $$2 \uparrow \uparrow n + 3 \uparrow \uparrow n$$ for any natural number $n \ge 5$. $2 \uparrow \uparrow n$ leaves the remainder ...
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### Fast converging sums involving tetrations

Loosely speaken, Liouville's theorem shows that rational series converging "too fast", have a transcendental limit. The concrete criterion is somewhat cumbersome and hard to check. Now my question : ...
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### Which precision would be needed?

According to Wikipedia, it is not known whether the number $$\pi \uparrow \uparrow 4$$ is an integer. (See Tetration) To which precision would $\pi$ have to be calculated to decide this ? The ...
### Is ${^5\pi}$ an integer? [duplicate]
Possible Duplicate: How to show $e^{e^{e^{79}}}$ is not an integer Is ${^5\pi}$ an integer? It is "obviously" not, right? But can we prove it? Here ${^5\pi}$ means the result of tetration ...