Leaving the specifics of prime numbers to others more capable than I, we know that while public-key encryption involves an asymmetric relationship between the public and the private key, there is a symmetric relationship between signing and encryption: we sign with our private key and encrypt with our recipient's public key, while verification is done with the sending party's public key and decryption with our private key.
Signing involves encrypting a checksum.
We now look at signing authorities, with a subset of them designated as trusted root authorities. As each authority has a public key, we may simply sign another's public key to indicate that we trust them. We thus form paths of trust.
The idea of cryptocurrency is to sign a transaction: the debit account always being the signing party. But the currency itself is merely a number verified by an algorithm--throw the book at me if you must--which is served on a first-come, first-served basis. To claim an unused number, all that is required is the ownership of two wallets.
The next thing is to con people into paying real money for your numbers: but this isn't the worst.
For, it requires a significant amount of processing to determine a valid, unused number. As the value of the currency increases, more and more electricity is turned to the task. Where electricity is cheap, there a datacentre will be.
And electricity is cheapest where pollution is not an official concern.
And without a One World Government, there are likely to remain, for the foreseeable future, places which cannot afford to concern themselves over pollution.
That cryptocurrency has become an institution can only be due to our obsession with technology; which obsession is not love. It is an attempt at outdoing those we inherited the tools from.
Simply for the bragging rights!