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 (that is, the product of the first n primes).
It follows that
The first few primorial primes are
As of 28 February 2012, the largest known primorial prime is 1098133# − 1 (n = 85586) with 476,311 digits, found by the PrimeGrid project.[1]
Euclid's proof of the infinitude of the prime numbers is commonly misinterpreted as defining the primorial primes, in the following manner: [2]
- 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; note that 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 cannot be a multiple of any of them).
See also
References
- A. Borning, "Some Results for
and 
" Math. Comput. 26 (1972): 567–570. - Chris Caldwell, The Top Twenty: Primorial at The Prime Pages.
- Weisstein, Eric W., "Primorial Prime", MathWorld.
- 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.
- ↑ Primegrid.com; forum announcement, 2 March 2011
- ↑ Michael Hardy and Catherine Woodgold, "Prime Simplicity", Mathematical Intelligencer, volume 31, number 4, fall 2009, pages 44–52.
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