Is 2(6)3 (2↑↑↑↑3) equal to 2^65536? And if yes, is 2(n)3 equal to 2 to the power of how many time 2 is repeated in the power tower?

Question

I am writing a paper for the last digits in a chain power of 2. I was wondering if 2↑↑↑↑3 is 2^65536. Beacouse 2↑↑↑3 is 65536 or 2^16 and is written as 2^2^...2^2 16 times and 2↑↑3 is 16 or 2^4 and is written as 2^...2^2 4 times.

Answer 1

We have

$$2\uparrow\uparrow\uparrow\uparrow 3=2\uparrow\uparrow\uparrow 2\uparrow\uparrow\uparrow 2=2\uparrow\uparrow\uparrow 4=2\uparrow\uparrow2\uparrow\uparrow\ 2\uparrow \uparrow 2=2\uparrow \uparrow 2\uparrow \uparrow 4=2\uparrow \uparrow 65536$$

So , your number is a power tower of $65536$ $2's$ , much larger than $2^{65536}$