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{{merge|Euclid number|date=February 2012}}
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{{morefootnotes|date=December 2011}}
 
In [[mathematics]], '''primorial primes''' are [[prime number]]s of the form ''p<sub>n</sub>''#&nbsp;±&nbsp;1, where ''p<sub>n</sub>''# is the [[primorial]] of ''p<sub>n</sub>'' (that is, the product of the first ''n'' primes).
 
It follows that{{Clarify|date=November 2013}}
 
: ''p<sub>n</sub>''# &minus; 1 is prime for ''n'' = 2, 3, 5, 6, 13, 24, ... {{OEIS|id=A057704}}
 
: ''p<sub>n</sub>''# + 1 is prime for ''n'' = 1, 2, 3, 4, 5, 11, ... {{OEIS|id=A014545}}
 
The first few primorial primes are
 
:[[3 (number)|3]], [[5 (number)|5]], [[7 (number)|7]], [[29 (number)|29]], [[31 (number)|31]], [[211 (number)|211]], 2309, 2311, 30029, 200560490131, 304250263527209
 
<!---{{As of|2010|url=http://primes.utm.edu/top20/page.php?id=5}}, the largest known primorial prime is 843301#&nbsp;&minus;&nbsp;1 with 365,851 digits, found in 2010 by the [[PrimeGrid]] project.<ref>[http://www.primegrid.com/download/prs-843301.pdf Primegrid.com]; official announcement, 24 December 2010</ref>--->
 
{{As of|2012|2|28|url=http://primes.utm.edu/top20/page.php?id=5}}, the largest known primorial prime is 1098133#&nbsp;&minus;&nbsp;1 (''n'' = 85586) with 476,311 digits, found by the [[PrimeGrid]] project.<ref>[http://www.primegrid.com/forum_thread.php?id=4146#51000 Primegrid.com]; forum announcement, 2 March 2011</ref>
 
[[Euclid]]'s [[proof (math)|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&ndash;52.</ref>
 
: Assume that the first ''n'' consecutive primes including [[2 (number)|2]] are the only primes that exist. If either ''p<sub>n</sub>''#&nbsp;+&nbsp;1 or ''p<sub>n</sub>''#&nbsp;&minus;&nbsp;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; note that each of these two numbers has a remainder of either ''p''&nbsp;&minus;&nbsp;1 or 1 when divided by any of the first ''n'' primes, and hence cannot be a multiple of any of them).
 
== See also ==
* [[Primorial]]
* [[Factorial prime]]
* [[Euclid number]]
* [[PrimeGrid]]
 
== References ==
* 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]].
* {{MathWorld|title=Primorial Prime|urlname=PrimorialPrime}}
* 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.
{{reflist}}
 
{{Prime number classes|state=collapsed}}
{{Num-stub}}
 
[[Category:Integer sequences]]
[[Category:Classes of prime numbers]]

Revision as of 04:12, 21 February 2014

Photography may be the cheapest way to avoid loss because of tragedy. Once your images have been in digital form, you can easily copy t...

Wedding photography is among the largest of wedding expenses. The photos are ready to tell the story of the wedding beautifully for decades to come. You wish to ensure you do everything you can to protect among the most critical memorial of your wedding day. You can simply take many steps to protect your wedding pictures.

Digital photography is the most inexpensive method to avoid loss as a result of tragedy. Once your photos are in digital form, it is possible to duplicate them. The digital technology enables fast backup and restore of your images. It is a technology that"s becoming very nearly inseparable from traditional photography. DVDs and burning CDs of one"s pictures is now quite popular, and it ought to be part of the photography company. You should go, if your photographer doesn"t offer digital copies.

The ink and paper applied are of outmost importance. What your images can look like decades once you get married greatly depends on the grade of the ink and paper. Ask to see what sort of guarantees the paper company provides. Ask your photographer to compare different sort of picture forms on the market.

No acids should be contained by your photo album. That includes the stuff, paper, and any material that is part of the scrapbook. Browse this web site source to research how to see this idea. You can contact the maker of the recording to be sure. Acids may cause discoloration over time.

Photograph photos must be laid flat on their backs. Standing photo albums can quicker bend. Be sure the wedding album includes guarantees you read and understand.

High temperatures and humidity are one of the worst enemies of a scrapbook. Most parts you would feel comfortable for prolonged periods are proper storage areas for your scrapbook. Learn further on a partner article directory by visiting mexico wedding photography. A warm attic during the summer or a wet and cold storage are great examples of bad storage places.

You have planned your wedding for a very long time, and you have spent a lot of money on your own photographs. Protect your investment!.

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