Qntm's Power Tower Toy

Fun fact with arrow notation, if you put it under a modulus it quickly converges to the same value no matter how high in exponents you go!

Eg. 2^2^2 = 2^4 mod 35 = 16

Let's go one higher

2^2^2^2 = 2^16 mod 35 = 16 too!

and once more for the record

2^2^2^2^2 = 2^65536 mod 35 = 16 as well. It'll keep giving this result no matter how high you go.

https://www.wolframalpha.com/input?i=2%5E2%5E2%5E2+mod+35 for a link of this to play with.

I could do this with any modulus and any exponent too.

2^3^3 = 2^3^3^3 = 7 mod 11 etc.

The reason is that the orders of powers are effected by the totient recursively and since totients always reduce, eventually the totient converges to 1. This is where the powers no longer matter under modulus. Eg. the totient of 35 is 12 (the effective modulo of the first order power), the totient of 12 is 2 (the effective modulo of the second order power), the totient of 2 is 1 (the effective modulo of the third order power) and so after 3 powers under mod 35 it converges.

I'm a big qntm fan. I highly recommend their "antimemetics" SCP stories and articles.

As someone from time to time peeking into googology.fandom.com , my favorite big number device probably still is loader.c, simply because of how concrete and unreachable it feels.

Too bad most Friedman's work has linkrotted by now...

Ah, for a second I hoped it is another novel.

If you haven’t read “There is no antimemetics division”, do it now. Easily one of the top science fiction out there.

However buy the Penguin books 2025 edition, not the self-published free one — that version has a meh ending and suffers from not having an editor.

Is it buggy for at least 2^(n)^2? It gives 4 for any n, but surely for example 2^^2 = 2^(2^2) != 4?

Hey he does good scifi too