To discover very high primes, e.g. some over 10,000 digits, one cannot merely look at random numbers. For there are some functions that commonly produce primes, although no function actually is guaranteed to produce primes. Some common types of primes are factorial primes, primorial primes, repunit primes and Mersenne primes. For all the lists of primes mentioned on this post, please visit the Prime Pages.
A factorial prime is a prime that is one more or one less than a factorial. A factorial, represented n!, is the product of all the counting numbers below and including n. For example, 3!=3*2*1=6, and 5!=5*4*3*2*1=120. Therefore, a factorial prime is one of the form n!+1 or n!-1. The first few factorial primes are 2 (1!+1), 3 (2!+1), 5 (3!-1), 7 (3!+1), 23 (4!-1), 719 (6!-1), 5039 (7!-1), 39916801 (11!+1), 479001599 (12!-1), and 87178291199 (14!-1). Currently, the highest factorial prime known is 34790!-1 which has 142,891 digits!
The next type of prime, the primorial prime, is a prime that is one more or one less than a primorial. Similar to a factorial a primorial (n#) is the product of all primes up to and possibly including n. For example, 11#=2*3*5*7*11=2310. The first few primorial primes are 3 (2#+1), 5 (3#-1), 7 (3#+1), 29 (5#-1), 31 (5#+1), 211 (7#+1), 2309 (11#-1), 2311 (11#+1), 30029 (13#-1), 200560490131 (31#+1), 304250263527209 (41#-1). The highest known primorial prime is 392113#+1, which has 169,966 digits.
The top eight largest primes known are all Mersenne primes. All Mersenne primes are of the form 2^n-1. It has been proven that 2^n-1 can only be prime if n is prime. The first few Mersenne primes are 3 (2^2-1), 7 (2^3-1), 31 (2^5-1), 127 (2^7-1), 8191 (2^13-1), 131071 (2^17-1), 524287 (2^19-1), 2147483647 (2^31-1) and 2305843009213693951 (2^61-1). In total, there are only 46 Mersenne primes in existence. The highest Mersenne prime known is 2^43112609-1, which has 12,978,189 digits. For all the Mersenne primes and their english names (e.g. 2,147,483,647= two billion, one hundred forty seven million, four hundred eighty three thousand, six hundred forty seven) see here. For the highest Mersenne prime in existence, the first -illion in the name (million, billion, trillion, quadrillion) is quattuormilliamilliatrecensexviginmilliaunsexagintillion (wow)!
The final type I will discuss here, the repunit prime, is, on my opinion, the most peculiar type of prime. A repunit is a number consisting of only ones (e.g. 111 or 11111111). Only a handful of this are repunit primes. A repunit is written is Rn with n ones (R3=111) The first few repunits that are prime (as a trivial case, 1 is excluded) are R2=11, R19=1111111111111111111, R23=11111111111111111111111, and R317=11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111. The highest repunit prime is only R1031, but there are many probable primes that are in the process of being proved prime.
For the 5000 highest primes, visit here.
A factorial prime is a prime that is one more or one less than a factorial. A factorial, represented n!, is the product of all the counting numbers below and including n. For example, 3!=3*2*1=6, and 5!=5*4*3*2*1=120. Therefore, a factorial prime is one of the form n!+1 or n!-1. The first few factorial primes are 2 (1!+1), 3 (2!+1), 5 (3!-1), 7 (3!+1), 23 (4!-1), 719 (6!-1), 5039 (7!-1), 39916801 (11!+1), 479001599 (12!-1), and 87178291199 (14!-1). Currently, the highest factorial prime known is 34790!-1 which has 142,891 digits!
The next type of prime, the primorial prime, is a prime that is one more or one less than a primorial. Similar to a factorial a primorial (n#) is the product of all primes up to and possibly including n. For example, 11#=2*3*5*7*11=2310. The first few primorial primes are 3 (2#+1), 5 (3#-1), 7 (3#+1), 29 (5#-1), 31 (5#+1), 211 (7#+1), 2309 (11#-1), 2311 (11#+1), 30029 (13#-1), 200560490131 (31#+1), 304250263527209 (41#-1). The highest known primorial prime is 392113#+1, which has 169,966 digits.
The top eight largest primes known are all Mersenne primes. All Mersenne primes are of the form 2^n-1. It has been proven that 2^n-1 can only be prime if n is prime. The first few Mersenne primes are 3 (2^2-1), 7 (2^3-1), 31 (2^5-1), 127 (2^7-1), 8191 (2^13-1), 131071 (2^17-1), 524287 (2^19-1), 2147483647 (2^31-1) and 2305843009213693951 (2^61-1). In total, there are only 46 Mersenne primes in existence. The highest Mersenne prime known is 2^43112609-1, which has 12,978,189 digits. For all the Mersenne primes and their english names (e.g. 2,147,483,647= two billion, one hundred forty seven million, four hundred eighty three thousand, six hundred forty seven) see here. For the highest Mersenne prime in existence, the first -illion in the name (million, billion, trillion, quadrillion) is quattuormilliamilliatrecensexviginmilliaunsexagintillion (wow)!
The final type I will discuss here, the repunit prime, is, on my opinion, the most peculiar type of prime. A repunit is a number consisting of only ones (e.g. 111 or 11111111). Only a handful of this are repunit primes. A repunit is written is Rn with n ones (R3=111) The first few repunits that are prime (as a trivial case, 1 is excluded) are R2=11, R19=1111111111111111111, R23=11111111111111111111111, and R317=11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111. The highest repunit prime is only R1031, but there are many probable primes that are in the process of being proved prime.
For the 5000 highest primes, visit here.
There are many other types of primes, only a few of which I described here.
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