Erdős problem 375

There exists a constant c > 0 such that for all n, if k < c * (log n / (log (log n))) ^ 3 → (∀ i < k, ¬ (n + i + 1).Prime), then there are distinct primes p₁, ... pₖ such that pᵢ ∣ n + i for all 1 ≤ i ≤ k. This is proved in [RST75]. There is no need to only consider sufficiently large n because one can always take c small enough so that k < c * (log n / (log (log n))) ^ 3 implies that k = 0 until n is large.