mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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 (mathematics), 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 ...

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 wheth ...

s show that: : ''p_n''# − 1 is prime for ''n'' = 2, 3, 5, 6, 13, 24, 66, 68, 167, 287, 310, 352, 564, 590, 620, 849, 1552, 1849, 67132, 85586, 234725, 334023, 435582, 446895, ... . (''p_n'' = 3, 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877, 843301, 1098133, 3267113, 4778027, 6354977, 6533299, ... ) : ''p_n''# + 1 is prime for ''n'' = 0, 1, 2, 3, 4, 5, 11, 75, 171, 172, 384, 457, 616, 643, 1391, 1613, 2122, 2647, 2673, 4413, 13494, 31260, 33237, 304723, 365071, 436504, 498865, ... . (''p_n'' = 1, 2, 3, 5, 7, 11, 31, 379, 1019, 1021, 2657, 3229, 4547, 4787, 11549, 13649, 18523, 23801, 24029, 42209, 145823, 366439, 392113, 4328927, 5256037, 6369619, 7351117, ..., ) The first term of the third sequence is 0 because ''p''₀# = 1 (we also let ''p''₀ = 1, see Primality of one , hence the first term of the fourth sequence is 1) is the

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

, and thus ''p''₀# + 1 = 2, which is prime. Similarly, the first term of the first sequence is not 1 (hence the first term of the second sequence is also not 2), because ''p''₁# = 2, and 2 − 1 = 1 is not prime. The first few primorial primes are 2, 3, 5, 7, 29, 31,

211 Year 211 ( CCXI) was a common year starting on Tuesday of the Julian calendar. At the time, in the Roman Empire it was known as the Year of the Consulship of Terentius and Bassus (or, less frequently, year 964 ''Ab urbe condita''). The denomin ...

, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 . , the largest known prime of the form ''p''_''n''# − 1 is 6533299# − 1 (''n'' = 446,895) with 2,835,864 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 7351117# + 1 (''n'' = 498,865) with 3,191,401 digits, also found by the PrimeGrid project.

Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...

proof Proof most often refers to: * Proof (truth), argument or sufficient evidence for the truth of a proposition * Alcohol proof, a measure of an alcoholic drink's strength Proof may also refer to: Mathematics and formal logic * Formal proof, a co ...

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 proven by Euclid in his work '' Elements''. There are several proofs of the theorem. Euclid's proof Euclid of ...

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