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Problem: Are there infinitely many primorial primes?

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:

  1. Are there infinitely many primes \(p\), for which \(p\downarrow+1\) is also prime?
  2. Are there infinitely many primes \(p\), for which \(p\downarrow+1\) is composite?
  3. Are there infinitely many primes \(p\), for which \(p\downarrow-1\) is also prime?
  4. Are there infinitely many primes \(p\), for which \(p\downarrow-1\) is composite?

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, respectively2.

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Footnotes

  1. The name "primorial of \(p\)" for the product of all primes \(\le p\) was introduced 1987 by Dubner, probably in accordance with the name "factorial" \(n!\), meaning the product of all natural numbers \(\le n\). ↩

  2. See Paolo Ribenboim, "The Little Book of Bigger Primes", Springer New York, 2004 ↩