What happens to primecoin if mathematicians discover some algorithm that would otherwise define the set of numbers that includes all primes?
It is impossible to find all prime numbers as there are an infinite number of them. All we can do is find them faster.
It is is possible that a fast algorithm (polynomial time) will be found for testing if a number is prime. This is considered to be extremely unlikely by the mathematical and computer science community. A fast algorithm for finding primes would break most public key cryptography systems in use today. Including the ones that are used for online shopping, https/ssl/tls, and online banking.
People have been studying it intensely for a long time now and finding such an algorithm would be a major mathematical achievement.
Quantum computers can do it but it’s not clear we will every be able to build a non trivial machine.
more reading here http://www.cs.virginia.edu/~kam6zx/is-it-secure/the-hardness-of-factoring/
Excellent post Lasle2, thank you for the link as well 
as there are an infinite number of themdoes not mean they all could not be found. Like if somebody figure out the prime number pattern and derive it into a formular, we could say that we have them all.
The most recent progress in this regard came from a previously unknown mathematician “Yitang Zhang” -
Unheralded Mathematician Bridges the Prime Gap.
In terms of the OP’s question: This is far from finding a formula that would derive all possible prime numbers. In fact it supports the theorem that there are an infinite number of primes, as the gap between two prime numbers in a chain will always be less than 70 million.