A New Postulate and Some Conjectures Concerning Pair Primes in the Interval [n!, (n + k)!]

Abstract

This paper rather studies the behaviour of prime numbers bounded below and above by positive integers n! and (n + k)!, and then after some numerical evidence, postulates that there is at least one pair primes of gap k ∈ 2Z⁺ in between n! and (n + k)! for every integer n ≥ 2 and every even integer k > 0. This assertion would eventually provide another structural form for Euclid theorem of ifinitude of primes, a kind of projection of the form in the original Bertrand postulate (now Chebychev's theorem). The truth- fulness of the conjecture that emanated from this postulate implies the Polignac's conjecture which aptly generalizes the twin prime conjecture. We thus present the new postulate and the conjectures for future research.