Primorial prime
Encyclopedia
In mathematics
, primorial primes are prime number
s of the form pn# ± 1, where:
The first few primorial primes are
, the largest known primorial prime is 843301# - 1 with 365,851 digits, found in 2010 by the PrimeGrid
project.
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 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).
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, primorial primes are prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s of the form pn# ± 1, where:
- pn# is the primorialPrimorialIn mathematics, and more particularly in number theory, primorial is a function from natural numbers to natural numbers similar to the factorial function, but rather than multiplying successive positive integers, only successive prime numbers are multiplied...
of pn (that is, the product of the first n primes).
- pn# − 1 is prime for n = 2, 3, 5, 6, 13, 24, ...
- pn# + 1 is prime for n = 1, 2, 3, 4, 5, 11, ...
The first few primorial primes are
- 3, 5, 7, 2929 (number)29 is the natural number following 28 and preceding 30.-In mathematics:It is the tenth prime number, and also the fourth primorial prime. It forms a twin prime pair with thirty-one, which is also a primorial prime. Twenty-nine is also the sixth Sophie Germain prime. It is also the sum of three...
, 3131 (number)31 is the natural number following 30 and preceding 32.- In mathematics :Thirty-one is the third Mersenne prime as well as the fourth primorial prime, and together with twenty-nine, another primorial prime, it comprises a twin prime. As a Mersenne prime, 31 is related to the perfect number 496,...
, 211211 (number)211 is the natural number between 210 and 212. It is also a prime number.-In mathematics:211 is an odd number.211 is a primorial prime, sum of three consecutive primes , Chen prime, centered decagonal prime, and self prime....
, 2309, 2311, 30029, 200560490131, 304250263527209
, the largest known primorial prime is 843301# - 1 with 365,851 digits, found in 2010 by the PrimeGrid
PrimeGrid
PrimeGrid is a distributed computing project for searching for prime numbers of world-record size. It makes use of the Berkeley Open Infrastructure for Network Computing platform...
project.
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 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).
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.