I have discovered the same thing while practicing prime sieve algorithms back in the day. Such properties of primes are quite useful for optimizing both speed and memory of sieve algorithms.

More generally, if you take the first n primes p_1, ..., p_n and define P=p_1*...*p_n, then all primes bigger than P be in the form of P*k +- a, such that a < P and GCD(P, a) = 1. In case of n=2 (P=2*3=6), there is this nice property that the only such a are 1 and 5 (which are equivalent), but in principle, the same can be done for any n. It's just that the set of all a has the size equal to Euler's totient function of P, which grows pretty fast as n increases.

For example, if n = 3, then P = 2*3*5 = 30, so all prime numbers bigger than 30 have to be in the form of 30k +- 1, 30k +- 7, 30k +- 11, 30k +- 13, 30k +- 17, 30k +- 19, 30k +- 23 or 30k +- 29 (notice that half of these are equivalent to the other half and can be ommitted). It is interesting that in this particular case, all a are either 1 or a prime less than P: I don't think that property holds for all n, though.