I’m Ray, and I’ve recently finished an article about the history of Proof-of-Work. I’ve tried to mention every important point, starting from pre-bitcoin functions and then delving into every algorithm that shifted the further development and adoption of cryptocurrencies, namely:

So, I thought it might be interesting for you to read it, since the algos of Peercoin & Primecoin were mentioned as well.

All this work is useless! All these hashes... no one needs them anywhere except cryptocurrencies. Well, you can, of course, print out and put on a wall the smallest discovered hashes, but they have no practical use. We are not going into disputes like 'should this work be useful or not', but we want to point out the fact that inventing a PoW function providing some benefit is an exotic task.

Still, such function was found. Developer named SunnyKing (he was the one to invent [or, at least, the first to implement] the Proof-of-stake scheme) presented the following idea: the proof of performed work is a discovered sequence of prime numbers conforming to some requirements. Or, rather, it should be Cunningham chain of the first or second kind (so called bi-twin primes).

Very interesting study you have done there and yes leads nicely to show the foresight of SunnyKing with Peercoin POS POW system and Primecoin’s generating of prime numbers.

the POS was not giving a name as far as I am aware… Sunny just refers to it as a POW POS hybrid as far as I am aware.

Very interesting study you have done there and yes leads nicely to show the foresight of SunnyKing with Peercoin POS POW system and Primecoin’s generating of prime numbers.

the POS was not giving a name as far as I am aware… Sunny just refers to it as a POW POS hybrid as far as I am aware.

[...]
Still, such function was found. Developer named SunnyKing (he was the one to invent [or, at least, the first to implement] the Proof-of-stake scheme) presented the following idea: the proof of performed work is a discovered sequence of prime numbers conforming to some requirements. Or, rather, it should be Cunningham chain of the first or second kind (so called bi-twin primes).

I’ll be happy to hear your comments and have a further discussion.[/quote]

Hi and welcome!

I haven’t visited the links bit I need to comment on the quote. Something got messed up there.
Proof-of-Stake is something completely different from PoW.
Peercoin is the first successful implementation of PoS (in a hybrid form that has a PoW process do distribute coins) and depending on the source invented by SunnyKing as well!
A layman explanation of PoS could say that in PoS the limited resource no longer is computing power, but the coins (which accumulate age as long as they keep unspent and can then mint a block by chance).

Back to Primecoin.
It’s right that this was invented and created by SunnyKing and the PoW indeed uses computing power to search of Cunningham Chains of the first and second kind as well as bi-twin chains (bi-twin chains are neither Cunningham Chains of the first nor the second kind, but a different kind; you may want to have a look here: https://en.wikipedia.org/wiki/Cunningham_chain). So this PoW does actually something meaningful for number theory. And just in case somebody did create energy efficient ASICS for the Primecoin PoW (don’t see that happen anytime soon) mathematicians wouldn’t be mad about that
So either way this PoW does something useful!