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: ''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 5 , 7 , 29 , 31 , 211, 2309, 2311, 30029 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 ''n''th 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). SEE ALSO EXTERNAL LINKS
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