##
**
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
*p* – *n*! 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*! + *r*^{2},
where *r* is the smallest prime larger
than a given natural number *n*, then *p* – *n*! 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*.

**
Full version: pdf,
dvi,
ps,
latex
**

(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.

Return to
**Journal of Integer Sequences home page**