# Comparison of $4$-entry-conway-chains and $3$-entry-conway-chains

• How big must $n$ approximately be, that

$$n\rightarrow n \rightarrow n\ \approx \ 3 \rightarrow 3 \rightarrow 3 \rightarrow 3$$

holds ?

$n\rightarrow n \rightarrow n$ (conway-chain) is equivalent to $n \uparrow^n n$ (Knuth's up-arrow notation)

• Is there a general method to calculate approximately $n$ such that

$$n \rightarrow n \rightarrow n \approx k \rightarrow k \rightarrow k \rightarrow k$$

for a given $k$ ?

• Be advised that Graham's number lies between $3\to 3\to 64 \to 2$ and $3\to 3 \to 65 \to 2$, while what you're asking for is $3 \to 3\to (3\to 3\to 27 \to 2) \to 2$, which is much larger. So the number $n$ will be quite large. – Arthur Nov 29 '14 at 19:06

$$n \approx 3 \rightarrow 3 \rightarrow ([3 \rightarrow 3 \rightarrow 27 \rightarrow 2] -1) \rightarrow 2$$
Since even $3 \rightarrow 3 \rightarrow 27 \rightarrow 2$ is quite large, this is approximately $3 \rightarrow 3 \rightarrow 3 \rightarrow 3$.
To see this, note that $$3 \rightarrow 3 \rightarrow 3 \rightarrow 3 = 3 \rightarrow 3 \rightarrow (3 \rightarrow 3 \rightarrow 27 \rightarrow 2) \rightarrow 2$$
And $$(a \rightarrow a \rightarrow b \rightarrow 2) \uparrow^{(a \rightarrow a \rightarrow b \rightarrow 2)} (a \rightarrow a \rightarrow b \rightarrow 2) \approx a \rightarrow a \rightarrow (b+1) \rightarrow 2$$
For larger $k$, one will have $n \approx k\rightarrow k\rightarrow k\rightarrow k$.