GRIMM’S CONJECTURE ON CONSECUTIVE INTEGERS (DEDICATED TO THE MEMORY OF PROFESSOR V. C. DUMIR)

Abstract. For positive integers n and k, it is possible to choose primes P1, P2, · · · , Pk such that Pi|(n+ i) for 1 ≤ i ≤ k whenever n+1, n+2, · · · , n+k are all composites and n ≤ 1.9 × 1010. This provides a numerical verification of Grimm’s Conjecture. Let n ≥ 0 and k ≥ 1 be integers. For an integer ν> 1, we denote by ω(ν) and P (ν) the number of distinct prime divisors of ν and the greatest prime factor of ν, respectively, and let ω(1) = 0, P (1) = 1. Let pi denote the i − th prime number. We shall always write p for a prime number. Let N0 = 8.5 × 108. We state a Conjecture of Grimm [2]. Suppose n + 1, · · · , n + k are all composite numbers and there are distinct primes Pi such that Pi|(n+ i) for 1 ≤ i ≤ k. Then we say that Grimm’s Conjecture holds for n and k. Further we say that Grimm’s Conjecture holds if there are distinct primes Pi such that Pi|(n+ i) for 1 ≤ i ≤ k whenever n+ 1, · · · , n+ k are all composites. If k ≥ n, it is well-known that the interval [n + 1, n + k] contains a prime. Thus k < n if n + 1, · · · , n + k are all composite numbers. According to Erdős (see [1]), this conjecture implies pi+1 − pi ≤ c1