Journal of Integer Sequences, Vol. 25 (2022), Article 22.1.4

## On Remarkable Properties of Primes Near Factorials and Primorials

### Antonín Čejchan Institute of Physics Czech Academy of Sciences Cukrovarnická 112/10 CZ – 162 00 Prague 6 Czech Republic Michal Křížek Mathematical Institute Czech Academy of Sciences Žitná 25 CZ – 115 67 Prague 1 Czech Republic Lawrence Somer Department of Mathematics Catholic University of America Washington, DC 20064 USA

Abstract:

The distribution of primes is quite irregular. However, it is conjectured that if p is the smallest prime greater than n! + 1, then pn! is also prime. We give a sufficient condition that guarantees when this conjecture is true. In particular, we prove that if a prime number p satisfies n! + 1 > p > n! + r2, where r is the smallest prime larger than a given natural number n, then pn! is also a prime. Similarly we treat another conjecture: If p is the largest prime smaller than n! – 1, then n! – p is also prime. Then we establish further sufficient conditions also for the case when n! is replaced by q#, which is the product of all primes not exceeding the prime q.

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(Concerned with sequences A005235 A033932 A035346 A037151 A037153 A037155 A046066 A055211 A087421 A098166 A098168.)

Received November 11 2021; revised version received January 3 2022; January 10 2022. Published in Journal of Integer Sequences, January 10 2022.