I can imagine an alternative crypto-currency based entirely on a general idea for a "useful" proof of work.  To me the most general form of proof of work consists of giving a proof of a mathematical theorem. It's hard to come up with a proof, but easy to check it is a proof. This standard asymmetry is important for the problems solved by proof of work schemes.

Now the thing that's really, I guess, different about this idea (from my understanding) is computers are still much worse than people at coming up with proofs.  This means people would be doing the "mining" and get "rewarded" by tokens. They could use the tokens to "mine" blocks.

This proposal seems like one that would be interesting for scientists and developers. But does anyone have the interest to fully flesh out the idea? Does anyone know of similar ideas or experiments out there?