Primorial prime
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In mathematics, primorial primes are prime numbers of the form pn# ± 1, where:
- pn# is the primorial of pn.
- pn# − 1 is prime for n = 2, 3, 5, 6, 13, 24, ... (Sloane A057704)
- pn# + 1 is prime for n = 1, 2, 3, 4, 5, 11, ... (Sloane A014545)
The first few primorial primes are
3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209
As of 2005, the largest known primorial prime is 392113#+1, found in 2001 by Daniel Heuer.
The idea of primorial primes appears in Euclid's proof of the infinitude of the prime numbers: First, assume that the first n primes are the only primes that exist. If either pn# + 1 or pn# - 1 is a primorial prime, it means that there are larger primes than the nth prime (if neither is a prime, that also proves the infinitude of primes, but less directly; note that each of these two numbers has a remainder of either p−1 or 1 when divided by any of the first n primes, and hence cannot be a multiple of any of them).
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