Anything with a use has value to a person benefiting from that use. The trick is to make it such that out-of-band payments to specific miners are not likely to occur.
That’s not to say that it’s absolutely, 100% a priori impossible to have a “useful but no-value” proof of work. Only theorized example I know of is time-lock decryption: you have a deterministic process for generating public keys, and the proof of work solution is the recovered private key. However this invalidates other assumptions that are required of a secure proof of work, at least with all existing public key crypto systems.
It needs to be valueless, not useless. Finding primes actually seems like something that might be both useful and low-value.
Anything with a use has value to a person benefiting from that use. The trick is to make it such that out-of-band payments to specific miners are not likely to occur.
That’s not to say that it’s absolutely, 100% a priori impossible to have a “useful but no-value” proof of work. Only theorized example I know of is time-lock decryption: you have a deterministic process for generating public keys, and the proof of work solution is the recovered private key. However this invalidates other assumptions that are required of a secure proof of work, at least with all existing public key crypto systems.