Thats it (afaik), and it could be for the usual, dismissive reasons, but its easy to hand-waive the "make primality a part of the work" part but it also comes down to the properties of the work that require it to be useful:
the difficulty of the work must be adjustable,
the difficulty/reward ratio must scale to the polynomial of users/work-rate to avoid sybil/"51% (31%)" attacks, and dissuade volatility during transitions
must be easily verifiable,
Primecoin uses Cunningham Chain primes - basically sequences of primes where 2x+1 is prime.
They are marginally useful with other applications on the horizon.
I could see adjusting the arbitrary rule-set - similar to the varying rulesets of cellular automata, like Conways - to further Number Theory/Game Theory/Swarm Economics at a general interdisciplinary level to be the most potentially rewarding, covering a larger swath of unknown unknowns.
My favorite “Practical POW” remains komoglorav complexity computation. The reward would likely scale with the runtime needed to verify a complexity, but there’s plenty of room for subtleties in the implementation. (for instance what happens when you prove a prior established complexity wrong?)
>(for instance what happens when you prove a prior established complexity wrong?)
what do you mean? you run their wallets, pun intended!
No stakes, no steaks!
But it does seem interesting - counterintuitive really, but a "Busy Beaver" / proof of work verifying mechanism enumerating inputs/instructions/outputs randomly (or whatever the nodes think they know best at ) while rewarding (only? why not top 3?) the shortest, most efficient block...could be tweaked to crunch ETH contracts like gas, brute-force fuzz-test legacy unsafe sourcecode...literally a foundation for further distributed computation.
There are languages like it - Dennis and his Bubblegum - that have generative, selective, and compressive patterns interned already.
Gapcoin (finding large gaps between successive primes)
Riecoin (finding maximally dense prime clusters of size 6)
Nexus (finding almost-dense clusters with a maximum spacing between successive primes)
As an aside, picking a mathematically interesting and intricate proof of work function is probably a bad idea, because someone like me will come along and optimize the miner and mine privately at a large profit margin, as I did with two of these coins.
I found this but curious what else exists! https://en.wikipedia.org/wiki/Primecoin