Maybe it would be possible to create a block-chain technology in which their participants calculate something scientifically meaningful.
Wikipedia lists a few https://en.wikipedia.org/wiki/List_of_distributed_computing_...
The impact may be small, but I see no reason why the impact should be there in the first place just to satisfy some mathematicians' curiosity about special numbers.
A more likely scenario for such independent instructions would operate on an entire bit string, like boolean operators and vector instructions. I think you'd have a tough time producing any useful output from such an algorithm, though, because you wouldn't be able to do much with conditionals to keep the branch predictor score fair.
I don't think that there are any algorithms that could operate within a generic benchmark that could have random elements in them _and_ product a useful result. Either the calculations are different and fair but meaningless, or they're exactly the same with the same result.
One that springs to mind is Monte-Carlo sampled raytracing. An individual ray might take more or less time to compute, but the time to compute 10 million rays will be statistically be roughly constant. You could even imagine averaging a bunch of renders of the same scene from different machines to get a lower-noise result, thereby demonstrating a benchmark combined with useful work.
Statistical predictability is the key.
(Confession - this isn't exactly theoretical. I sometimes have occasion to render light fields, and I shard the work over as many random workstations as I can get my hands on. It's always obvious which workstations are faster than others, even without making any special attempt to balance the workload. I think this is actually a workable concept.)
Which now that I think about it, of course it would be like that. But still, what a headache for someone who just wants The Best One.
Tape measures are useful even if they're not the same length.
Higher-end engineering tapes acknowledge that length isn't constant, and provide temperature coefficients, ranges of accuracy, and tensioning guidelines.
You can spend a lot of money on a tape measure which is more exact under more conditions. Most people don't need this.
It proves it because with the selected algorithm, there's no known shortcut to just doing a load of hashes, so if you found one it means you did a lot of hashing.
What problem would you insert instead that had the same properties?
We should also not forget that it gets harder and harder for the same benefit. Any amount of energy can be poured into a PoW system. Invent fusion, use it all in crypto.
The problem then is more economics, who has an incentive to discover more Mersenne primes?
With a PoW that has "external rewards", the cost of attacking the chain falls in proportion to the external reward. You are correct that it does not fall to zero.
No individual would pay for it, but it’s still more valuable than random noise.
Now one can argue that mathematical patents provide more costs than benefits, which is a line of debate I am certainly open to. End that, and math research would be a public good.
But in the system we have now, a huge amount of mathematical research has generated a significant amount of money for very many people, from the RSA algorithm to Penrose patenting his tilings (and getting money from toilet paper makers that used his aperiodic tilings to make their TP more fluffy [1])
What I think you mean to say is that the market consistently undervalues mathematical research. Unfortunately it's not so easy to actually demonstrate, because given the huge amount of math research produced, only a very small amount of that ends up being useful in the future and often the original discoverer is long dead. This is similar to the situation with other arts and sciences. A small number of people drive the field forward and the vast majority of scientific publications are basically filler. Certainly the world would be better with more effective funding mechanisms that we have on offer, which is basically a game of capture-the-grant.
[1]https://apnews.com/article/59354892d1875c277c91afd3fb5ca516
You can use these distributed computing projects for burn-in testing though. When you just want to run your computer hard to make sure it can manage N hours at 100%, and don't necessarily need to know how it compares to others, distributed computing can work (although, there's some risk of wrong results and therefore more work for the organizers if you're testing your system in extreme conditions, sometimes those lead to miscalculations)