Sets of Prime Numbers Satisfying a Divisibility Condition Original Research Article

We study setsPofkprimes that satisfy the condition gcd(∏ A−∏ B, ∏ P)=1 wheneverAandBare disjoint non-empty subsets ofP. It is known that such sets of primes exist for all positive integers k.It is of interest to know the asymptotic behavior ofnk, the smallest natural number that is the product of k such primes. In this paper we derive asymptotic bounds fornk.