Primorial prime
Prime number that is product of first n primes ± 1
In mathematics, a primorial prime is a prime number of the form pn# ± 1, where pn# is the primorial of pn (i.e. the product of the first n primes).[1]
Primality tests show that:
- pn# − 1 is prime for n = 2, 3, 5, 6, 13, 24, ... (sequence A057704 in the OEIS).
- pn# + 1 is prime for n = 0, 1, 2, 3, 4, 5, 11, ... (sequence A014545 in the OEIS).
The first term of the second sequence is 0 because p0# = 1 is the empty product, and thus p0# + 1 = 2, which is prime. Similarly, the first term of the first sequence is not 1, because p1# = 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 (sequence A228486 in the OEIS).
As of October 2021[ref], the largest known primorial prime (of the form pn# − 1) is 3267113# − 1 (n = 234,725) with 1,418,398 digits, found by the PrimeGrid project.[2][3]
As of 2022[update], the largest known prime of the form pn# + 1 is 392113# + 1 (n = 33,237) with 169,966 digits, found in 2001 by Daniel Heuer.
Euclid's proof of the infinitude of the prime numbers is commonly misinterpreted as defining the primorial primes, in the following manner:[4]
- Assume that the first n consecutive primes including 2 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; 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 pn).
See also
- Compositorial
- Euclid number
- Factorial prime
References
- ^ Weisstein, Eric. "Primorial Prime". MathWorld. Wolfram. Retrieved 18 March 2015.
- ^ Primegrid.com; forum announcement, 7 December 2021
- ^ Caldwell, Chris K., The Top Twenty: Primorial (the Prime Pages)
- ^ Michael Hardy and Catherine Woodgold, "Prime Simplicity", Mathematical Intelligencer, volume 31, number 4, fall 2009, pages 44–52.
See also
- A. Borning, "Some Results for and " Math. Comput. 26 (1972): 567–570.
- Chris Caldwell, 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.
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Prime number classes
- Fermat (22n + 1)
- Mersenne (2p − 1)
- Double Mersenne (22p−1 − 1)
- Wagstaff (2p + 1)/3
- Proth (k·2n + 1)
- Factorial (n! ± 1)
- Primorial (pn# ± 1)
- Euclid (pn# + 1)
- Pythagorean (4n + 1)
- Pierpont (2m·3n + 1)
- Quartan (x4 + y4)
- Solinas (2m ± 2n ± 1)
- Cullen (n·2n + 1)
- Woodall (n·2n − 1)
- Cuban (x3 − y3)/(x − y)
- Leyland (xy + yx)
- Thabit (3·2n − 1)
- Williams ((b−1)·bn − 1)
- Mills (⌊A3n⌋)
- Fibonacci
- Lucas
- Pell
- Newman–Shanks–Williams
- Perrin
- Palindromic
- Emirp
- Repunit (10n − 1)/9
- Permutable
- Circular
- Truncatable
- Minimal
- Delicate
- Primeval
- Full reptend
- Unique
- Happy
- Self
- Smarandache–Wellin
- Strobogrammatic
- Dihedral
- Tetradic
- Twin (p, p + 2)
- Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …)
- Triplet (p, p + 2 or p + 4, p + 6)
- Quadruplet (p, p + 2, p + 6, p + 8)
- k-tuple
- Cousin (p, p + 4)
- Sexy (p, p + 6)
- Chen
- Sophie Germain/Safe (p, 2p + 1)
- Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...)
- Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...)
- Balanced (consecutive p − n, p, p + n)
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