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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.

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.

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

| | | | created: 2014-03-08 15:33:24 | modified: 2019-03-16 21:38:20 | by: bookofproofs

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