I would doubt it's possible, considering the methods that we use to detect primes come down to brute force. It's hard to build an algorithm when we don't have a formal proof of how primes work/can be detected.
Modern prime testing is quite far from brute force. Tests such as AKS[1], ECPP[2], and APR[3] can determine whether a number is prime in polynomial time (in the number of bits/digits), whereas a true brute force search would be exponential.
You're right, I misspoke I should have been more precise and said that out way of finding primes isn't based on a specific theorem or algorithm in that we still have to take a number and demonstrate its primeness.
Well it's definitely possible unless I'm a moron (correct me if I am). If you have the primes up to √n you can find all primes up to n because everything else will be composite of the primes you know already. This is basically what TFA is doing. What I was suggesting is that it would be cool if you could avoid defining rules explicitly for (3n+6), (5n+10), etc and have them appear from some feedback loop where we go from a sequence of primes up to n, to n^2, to n^4...
[1]https://en.wikipedia.org/wiki/AKS_primality_test
[2]https://en.wikipedia.org/wiki/Elliptic_curve_primality_provi...
[3]https://en.wikipedia.org/wiki/Adleman%E2%80%93Pomerance%E2%8...