depends on your definition of patterns..

Eratosthenes sieve is a pattern of sorts, but one with many dependent variables that we are unsure how to make agnostic from the pattern back to its initial state.. essentially we are unable to find the nth prime number, but if we already ran the pattern up to the nth prime number, finding n+1 is possible with enough effort

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mersenne primes are primes that all share the form of 2^p-1 and they appear all through out the momentum of prime numbers, but again, our problem is finding when the next one will occur and our best solutions boil down to rote testing..

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one person involved in the discovery here reported has a theorem named after a colleague and himself that is more along a traditional pattern, they call it an arithmetic progression, here is one such example..

    6,171,054,912,832,631 + 366,384 · 223,092,870 · n, for n = 0 to 24

each of the 24 numbers in that sequence will be prime

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i came across this momentum sometime last week:

   n**2-n+41

the first 40 values for n, 1..41, will produce primes.. after that the distribution of primes for whole values of n becomes erratic.. why? unsure and investigation quickly proved a difficult avenue of research so i let that distraction lie for another day

(i) http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes

(ii) http://en.wikipedia.org/wiki/Mersenne_prime

(iii) http://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem

There is Ulam's Sprial for example.

http://en.wikipedia.org/wiki/Ulam_spiral

Prime is the pattern?