When Doesn’t the Interval [an+k + Pn, an+k+1 + Pn] (n, k 2 N) Contain any Primes? Theorem and Counterexamples (an is the nth Prime Number & Pn is the nth Prime Factorial)

Hayat Rezgui

The objective of this short paper is to give and prove a main theorem confirming that any interval of the special form: ]an+k + Pn, an+k+1 + Pn[ (n 2 N) does not contain any primes, for all k 2 N such that an+k+1 < a2 n+1 (an is the nth prime number & Pn is the nth prime factorial). Then ...