Primorial prime
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), 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), 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". Wolfram. http://mathworld.wolfram.com/PrimorialPrime.html.
- ↑ 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 [math]\displaystyle{ k! + 1 }[/math] and [math]\displaystyle{ 2 \cdot 3 \cdot 5 \cdot p + 1 }[/math]" 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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