Number of prime divisors in a product of terms of an arithmetic progression

Abstract

Let d ≥ 1, k ≥ 3, n ≥ 1 be integers with gcd(n, d) = 1. We denote ∆ = ∆(n, d, k) = n(n+ d) · · · (n+ (k − 1)d). For an integer ν > 1, we write ω(ν) and P (ν) for the number of distinct prime divisors of ν and the greatest prime factor of ν, respectively. Further we put ω(1) = 0 and P (1) = 1. For l coprime to d, we write π(ν, d, l) for the number of primes ≤ ν and congruent to l modulo d. Further, we denote by πd(ν) for the number of primes ≤ ν and coprime to d. The letter p always denote a prime number. Let W (∆) denote the number of terms in ∆ divisible by a prime > k. We observe that every prime exceeding k divides at most one term of ∆. Therefore we have W (∆) ≤ ω(∆)− πd(k). (1) If max(n, d) ≤ k, we see that n+(k−1)d ≤ k and therefore no term of ∆ is divisible by more than one prime exceeding k. Thus W (∆) = ω(∆)− πd(k) if max(n, d) ≤ k. (2)