According to my understanding, generating an ever-growing list of prime numbers benefits mathematics and society in some mysterious way. Given this, why have more recently created proof-of-work focused coins largely ignored Primecoin's mining algorithm in favor of Bitcoin's, which yields no epiphenomenal benefits?
closed as primarily opinion-based by JBaczuk, Raghav Sood, Andrew Chow♦ Oct 3 '18 at 17:39
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One of the most important functions of mining is to secure the network, meaning that a user can be reasonably sure that their transactions are immutably recorded in the chain. With this in mind, it becomes very important that the game theory of mining incentivizes the miners to remain in consensus, and behave in the expected, 'honest' way.
In a simplified sense, this means that the cost of 'behaving well' (ie the way users want miners to behave, staying in consensus and securing the blockchain) must be less than the cost of 'behaving badly' (ie causing forks, withholding blocks, reorgs, etc). Likewise, the benefits of behaving well should outweigh the benefits of behaving badly.
why have more recently created proof-of-work focused coins largely ignored Primecoin's mining algorithm in favor of Bitcoin's, which yields no epiphenomenal benefits?
There is simple answer: if the hardware and algorithm used for mining provide external benefit, then that hardware will have value outside of it's use as a cryptocurrency miner. Thus, if a miner behaves badly and attacks the chain, their hardware will still have value, even if the chain dies. This lowers the cost of behaving badly, compared to a cryptocurrency which uses an otherwise useless mining algorithm. The net result is that the blockchain has a lower security guarantee, because the miners are not as invested in maintaining the chain.
Currently, we see that Bitcoin miners are spending huge amounts of energy to secure the network. The naïve understanding is that this is wasteful, the nuanced understanding is that Bitcoin itself benefits humankind in ways that make this cost worth it.
For more information related to this, see this question.