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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.
It is widely believed, but false, that 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).
In fact, Euclid's proof did not assume that a finite set contains all primes that exist. Rather, it said: consider any finite set of primes (not necessarily the first ''n'' primes; e.g. it could have been the set {3, ,11, 47}), and then went on from there to the conclusion that at least one prime exists that is not in that set. [1]
★ Factorial prime
★ Euclid number
★ The Prime Pages - The Top Twenty Primorial
★ Coordinated Search for Primorial Primes
: ''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.
It is widely believed, but false, that 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).
In fact, Euclid's proof did not assume that a finite set contains all primes that exist. Rather, it said: consider any finite set of primes (not necessarily the first ''n'' primes; e.g. it could have been the set {3, ,11, 47}), and then went on from there to the conclusion that at least one prime exists that is not in that set. [1]
| Contents |
| See also |
| External links |
See also
★ Factorial prime
★ Euclid number
External links
★ The Prime Pages - The Top Twenty Primorial
★ Coordinated Search for Primorial Primes
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