The distribution of primes is quite irregular. We prove the following theorem which however allows some regularity: If a prime number p satisfies n! + 1 < p < n! + r^2 , where r is the smallest prime larger than a given natural number n, then p - n! is also a prime. We state a similar theorem for primes just below n! - 1. Further we prove similar statements also for the case when n! is replaced by q# which is the product of all primes not exceeding a prime q.