We call the number \(p_r\downarrow:=p_1p_2\cdots p_r\) the primorial of the first \(r\) consecutive prime numbers smaller or equal \(p_r\)^{1}.
The Euclidean proof for the infinite number of primes provides no indication of how to find the next prime number \(P\). It only indicates that \(p_r < P\le p_r\downarrow+1\). Thus, for some indices \(r\), the number \(p_r\downarrow+1\) itself is a prime number, and for other indices it is composite.
The following problems are unsolved:
We call the numbers \(p\downarrow+1\) and \(p\downarrow-1\) primorial primes if they are prime.
In 2002, Galdwell & Gallot tested the primality of \(p\downarrow+1\) for primes \(p\) < 120000 and discovered that these numbers are prime only for the prime numbers \(p=\) 2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4587, 11549, 13649, 18523, 23801, 24029, and 42209. The corresponding results for \(p\downarrow-1\) were \(p=\) 3, 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, and 15877. Up to now, these results were confirmed for \(p < \) 637000 and \(p < \) 650000, respectively^{2}.