Typically, Wilson's is given as

$$(n-1)! \equiv -1 \pmod{n},$$

which is short and sweet, but I came up with an alternate presentation of it that's arguably needlessly complicated, but also arguably illustrative:

$$\prod_{i=1}^{n}i \equiv \sum_{i=1}^{n}{1} \pmod{\sum_{i=1}^{n}{i}}.$$

This shows pretty clearly how, in one sense, primes are an imbalance in the foot race between a product $1\cdot2\cdot3\cdots k$ and a sum $1+2+3+\dots+k$.

Thinking along these lines, I noticed that represented this way, you essentially have a mechanism that's counting constant units (the center sum), a mechanism that's counting constant groups (the mod sum), and a mechanism that's counting multiplicative groups of increasing size, or however you'd like to word it.

This jumped out at me as the first three levels of what are sometimes called hyperoperations, where the successor function$-$in this case, our constant $1$ sum$-$is usually the $0$th-level or base hyperoperation. There's a logical progression from there, with addition being level $1$, multiplication being level $2$, exponentiation being level $3$, tetration being level $4$, and so forth, as each higher level is in some sense a repetition of the previous operation.

That said, Wilson's as given above looks very much like primality could be related to the interplay between the first three hyperoperation levels, and now I'm wondering whether there is any way to incorporate the next one (exponentation) in a meaningful or useful way.

I experimented for a while along these lines, but essentially got nowhere. There were lots of factors popping out of places, but no patterns I could discern. There's also the question of implementation: I figured the most likely expression to fit in with these other three would be $2^{n!}$ or a minor variation like $2^{(n-1)!}$, so that's what I played with, but there are certainly other possibilities.

I did run into this sequence (A320886) at one point, which has something to do with integer partitioning, which made some sense given the context but wasn't immediately helpful. I suppose my question is whether this line of thinking seems compelling or seems like a waste of time, or if anyone knows of a situation in which $2^{n!}$ or similar arises, especially if it could potentially relate to this, I'd like to hear about it.

  • $\begingroup$ The tag wilsons-theorem is perhaps too specific. $\endgroup$ – Travis Willse Dec 5 '19 at 10:04
  • $\begingroup$ +1, because I think the insight is interesting. Nevertheless, I'd like to warn that hyperoperations are a rabbit hole that rarely leads anywhere. There's a reason tetration is barely heard of: it just doesn't have many interesting properties. Therefore, I doubt that there's a good way to answer your question. $\endgroup$ – URL Dec 7 '19 at 6:15
  • $\begingroup$ Point taken. That said, there was one thing I (re-re-re-re-re-) discovered about tetration that blew my mind at first: the fact that tetration of $a \pmod{m}$ repeats with a fixed period even for exponent towers of infinite height. It makes sense in retrospect, but still, that rabbit hole was worth it. $\endgroup$ – Trevor Dec 7 '19 at 10:37

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