The distribution of primes is quite irregular. However, it is conjectured that if p is the smallest prime greater than n! + 1, then p – n! is also prime. We give a sufficient condition that guarantees when this conjecture is true. In particular, we prove that if a prime number p satisfies n! + 1 > p > n! + r², where r is the smallest prime larger than a given natural number n, then p – n! is also a prime. Similarly we treat another conjecture: If p is the largest prime smaller than n! – 1, then n! – p is also prime. Then we establish further sufficient conditions also for the case when n! is replaced by q#, which is the product of all primes not exceeding the prime q.