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.
A classic would be quickly computing such big numbers under a modulus. You just compute the carmichael totient recursively till it hits 1, disregard higher orders and then going backwards calculate the powers, reducing by the modulo of the current order (this way it never gets large enough to be a pain to calculate). The totients reduce in logn time and each step is logn so it’s merely logn^2 to calculate.
There's a new, professionally-published book version of "There Is No Antimemetics Division" out as well[1], if you want to support Sam's work that way. I have print copies of both the self-published V1 and the new V2. I'm very excited about the latter, though I haven't finished it yet.
One small word of caution if you read the older version first: for what I assume are copyright reasons around using SCP in a professionally-published book, the new published version has had to strip out all the SCP references and change the names of all the characters, but it is otherwise very close to the old one. There are a handful of new scenes and some other small differences, but many pages and chapters are word-for-word identical apart from the aforementioned name changes.
This could just be a me thing, but I found this incredibly distracting after being so used to the old version, and just couldn't manage to enjoy it. Fortunately I bought the old one as well.
I’ve read the older version and really liked it, strange ending and all, and I’ve gifted the new version for X-mas. My xmas wish list is for a 6 episode mini-series funded by the fruit company.
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...
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.
[1]: https://qntm.org/antimemetics
This could just be a me thing, but I found this incredibly distracting after being so used to the old version, and just couldn't manage to enjoy it. Fortunately I bought the old one as well.
[0] https://qntm.org/mmacevedo
Too bad most Friedman's work has linkrotted by now...
This will make more sense if you look at how the inputs a,b,n in the toy (2,2,3) and (2,3,3) present differently.
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.