Primorial prime

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In mathematics, primorial primes are prime numbers of the form pn# ± 1, where pn# is the primorial of pn (the product of the first n primes). [1]

According to this definition,

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, as 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

As of 28 February 2012, the largest known primorial prime is 1098133# − 1 (n = 85586) with 476,311 digits, found by the PrimeGrid project.[2]

Euclid's proof of the infinitude of the prime numbers is commonly misinterpreted as defining the primorial primes, in the following manner: [3]

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[edit]

References[edit]

  1. ^ Weisstein, Eric. "Primorial Prime". MathWorld. Wolfram. Retrieved 18 March 2015. 
  2. ^ Primegrid.com; forum announcement, 2 March 2011
  3. ^ Michael Hardy and Catherine Woodgold, "Prime Simplicity", Mathematical Intelligencer, volume 31, number 4, fall 2009, pages 44–52.

See also[edit]

  • 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.