mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, a primorial prime is a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

of the form ''p_n''# ± 1, where ''p_n''# is the

primorial In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function ...

of ''p_n'' (i.e. the product of the first ''n'' primes).

Primality test A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whet ...

s show that : ''p_n''# − 1 is prime for ''n'' = 2, 3, 5, 6, 13, 24, ... : ''p_n''# + 1 is prime for ''n'' = 0, 1, 2, 3, 4, 5, 11, ... The first term of the second sequence is 0 because ''p''₀# = 1 is the

empty product In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors. It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question ...

, and thus ''p''₀# + 1 = 2, which is prime. Similarly, the first term of the first sequence is not 1, because ''p''₁# = 2, and 2 − 1 = 1 is not prime. The first few primorial primes are : 2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 , the largest known primorial prime (of the form ''p''_''n''# − 1) is 3267113# − 1 (''n'' = 234,725) with 1,418,398 digits, found by the

PrimeGrid PrimeGrid is a volunteer computing project that searches for very large (up to world-record size) prime numbers whilst also aiming to solve long-standing mathematical conjectures. It uses the Berkeley Open Infrastructure for Network Computing ...

project. , the largest known prime of the form ''p''_''n''# + 1 is 392113# + 1 (''n'' = 33,237) with 169,966 digits, found in 2001 by Daniel Heuer.

Euclid Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...

's proof of the

infinitude of the prime numbers Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proved by Euclid in his work '' Elements''. There are several proofs of the theorem. Euclid's proof Euclid offere ...

is commonly misinterpreted as defining the primorial primes, in the following manner:Michael Hardy and Catherine Woodgold, "Prime Simplicity", ''

Mathematical Intelligencer ''The Mathematical Intelligencer'' is a mathematical journal published by Springer Verlag that aims at a conversational and scholarly tone, rather than the technical and specialist tone more common among academic journals. Volumes are released qua ...

'', volume 31, number 4, fall 2009, pages 44–52. : Assume that the first ''n'' consecutive primes including 2 are the only primes that exist. If either ''p_n''# + 1 or ''p_n''# − 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; 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 all its prime factors are larger than ''p''_''n'').

See also

References

See also