This is Joseph Van Name. I have a Ph.D. in mathematics, and I have been the only consistent researcher of Laver tables for the past few years.
And you choose to make a challenging post to an online SIG - you seem to be spoiling for an intellectual dust-up - or at least a fine bonfire of straw men
Gapcoin, Riecoin, and Primecoin are all cryptocurrencies with useful mining algorithms. But are these mining algorithms really that useful and are these mining algorithms really that efficient at producing scientific research?
Openly and clearly
not --- because the application's primary purpose is, and always has been, to act as a cryptocurrency.
You do not see professional mathematicians writing programs to compute the next prime number.
You seem to be unaware of the work of the
mersenne forum. OTOH, perhaps you're not unaware of them but your research has ruled them out as they universally fail to pass your undefined (and ill-conceived) criterion of "professional mathematician".
Therefore, mining algorithms have quite severe limitations to how they can be set up to solve important computational problems. Therefore a prime-based mining algorithm will therefore have an extremely narrow scope of research advancement.
In which, despite the liberal use of "therefore", the characterisations "quite severe", "important" and "extremely narrow" are baldly presented as unsupported opinion or even casual conjecture. As ever, you need to show your workings-out.
Have any mathematicians used the results of cryptocurrency mining to prove things or further research prime numbers in any way?
You seem to misunderstand Gapcoin's approach - the proof-of-work is based on searching for
prime gaps of superior merit and Gapcoin currently holds the world record for the best merit, as published by the late Dr Tom Nicely of Lynchburg U a "professional mathematician" who in his time curated the
list of first occurrence prime gaps, the list now maintained by a community of interested mathematicians (I cannot vouch for their professional status, alas)
https://primegap-list-project.github.io/ Do the results of mining give any insight into number theory?
For Gapcoin, yes - it holds the world record (
https://en.wikipedia.org/wiki/Prime_gaps). For further information, see the
Prime Gap Search Group section of the Mersenne Forum.
These prime based mining algorithms are also a bit inefficient.
Well
obviously. It's a nuanced use of a proof-of-work calculation that is less wasteful that pure hashing. It's never pretended to be anything else.
Suppose that one has determined whether a number x is prime or not. ... the mining algorithm does not reward anyone for testing whether it is a part of a Cunningham chain.
Not relevant to Gapcoin's prime gap PoW.
So naturally, I would want to make a cryptocurrency with a Laver table based mining algorithm, but I decided not to pursue that area of research because I know that
Let me add another, courtesy of your good self:
These mining algorithms have some mathematical use ... but their use is quite limited.
So, what was your purpose in posting this here?
Cheers
Graham