{{Short description|Prime number that is product of first n primes ± 1}} {{Infobox integer sequence | terms_number = 52 | con_number = Infinite | parentsequence = ''p''# ± 1 | first_terms = 2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 | largest_known_term = 9562633# + 1 | OEIS = A228486 }} In mathematics, a '''primorial prime''' is a prime number of the form ''p<sub>n</sub>''# ± 1, where ''p<sub>n</sub>''# is the primorial of ''p<sub>n</sub>'' (i.e. the product of the first ''n'' primes).<ref>{{cite web|last1=Weisstein|first1=Eric|title=Primorial Prime|url=http://mathworld.wolfram.com/PrimorialPrime.html|website=MathWorld|publisher=Wolfram|access-date=18 March 2015|ref=3}}</ref>
Primality tests show that:
: ''p<sub>n</sub>''# − 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, ... {{OEIS|id=A057704}}. (''p<sub>n</sub>'' = 3, 5, 11, 13, 41, 89, 317, 337, 991, 1873, 2053, 2377, 4093, 4297, 4583, 6569, 13033, 15877, 843301, 1098133, 3267113, 4778027, 6354977, 6533299, ... {{OEIS|id=A006794}}) : ''p<sub>n</sub>''# + 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, ... {{OEIS|id=A014545}}. (''p<sub>n</sub>'' = 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, 9562633, ..., {{OEIS|id=A005234}})
The first term of the third sequence is 0 because ''p''<sub>0</sub># = 1 (we also let ''p''<sub>0</sub> = 1, see Primality of one, hence the first term of the fourth sequence is 1) is the empty product, and thus ''p''<sub>0</sub># + 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''<sub>1</sub># = 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 {{OEIS|id=A228486}}.
{{As of|2025}}, it is not known whether there are infinitely many primorial primes, and it is also not known whether infinitely many numbers of the form ''p<sub>n</sub>''# ± 1 are composite numbers.<ref>{{MathWorld |id=PrimorialPrime |title=Primorial Prime |access-date=2025-11-29}}</ref>
{{As of|2025|7}}, the largest known prime of the form ''p''<sub>''n''</sub># − 1 is 6533299# − 1 (''n'' = 446,895) with 2,835,864 digits, found by the PrimeGrid project.<ref name="t5k">https://t5k.org/top20/page.php?id=5#records</ref>
{{As of|2025|7}}, the largest known prime of the form ''p''<sub>''n''</sub># + 1 is 9562633# + 1 (''n'' = 637,491) with 4,151,498 digits, also found by the PrimeGrid project.<ref name="t5k"/>
Euclid's proof of the infinitude of the prime numbers is commonly misinterpreted as defining the primorial primes, in the following manner:<ref>Michael Hardy and Catherine Woodgold, "Prime Simplicity", ''Mathematical Intelligencer'', volume 31, number 4, fall 2009, pages 44–52.</ref>
: Assume that the first ''n'' consecutive primes including 2 are the only primes that exist. If either ''p<sub>n</sub>''# + 1 or ''p<sub>n</sub>''# − 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''<sub>''n''</sub>).
== See also == * Compositorial * Euclid number * Factorial prime
== References == {{reflist}}
== See also == * A. Borning, "Some Results for <math>k! + 1</math> and <math>2 \cdot 3 \cdot 5 \cdot p + 1</math>" ''Math. Comput.'' '''26''' (1972): 567–570. * Chris Caldwell, [http://primes.utm.edu/top20/page.php?id=5 ''The Top Twenty: Primorial''] at The Prime Pages. * Harvey Dubner, "Factorial and Primorial Primes." ''J. Rec. Math.'' '''19''' (1987): 197–203. * Paulo Ribenboim, ''The New Book of Prime Number Records''. New York: Springer-Verlag (1989): 4.
{{Prime number classes|state=collapsed}} {{Num-stub}}
Category:Integer sequences Category:Classes of prime numbers