Cauchy horizon: Difference between revisions

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{{Infobox integer sequence
| named_after          = [[Samuel S. Wagstaff, Jr.]]
| publication_year      = 1989<ref>{{Cite journal | last1 = Bateman | first1 = P. T. | author1-link = Paul T. Bateman | last2 = Selfridge | first2 = J. L. | author2-link = John Selfridge | last3 = Wagstaff, Jr. | first3 = S. S. | title = The New Mersenne Conjecture | journal = American Mathematical Monthly | year = 1989 | volume = 96 | pages = 125–128 | jstor = 2323195 }}</ref>
| author                = [[Paul T. Bateman|Bateman, P. T.]], [[John Selfridge|Selfridge, J. L.]], [[Samuel S. Wagstaff, Jr.|Wagstaff Jr., S. S.]]
| terms_number          = 43
| first_terms          = [[3 (number)|3]], [[11 (number)|11]], [[43 (number)|43]], [[683 (number)|683]]
| largest_known_term    = (2<sup>13372531</sup>+1)/3
| OEIS                  = A000979
}}
In [[number theory]], a '''Wagstaff prime''' is a [[prime number]] ''p'' of the form


:<math>p={{2^q+1}\over 3}</math>


where ''q'' is another prime. Wagstaff primes are named after the [[mathematician]] [[Samuel S. Wagstaff Jr.]]; the [[prime pages]] credit François Morain for naming them in a lecture at the Eurocrypt 1990 conference. Wagstaff primes are related to the [[Mersenne conjectures#New Mersenne conjecture|New Mersenne conjecture]] and have applications in [[cryptology]].
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The first three Wagstaff primes are 3, 11, and 43 because
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:<math>
\begin{align}
3 & = {2^3+1 \over 3}, \\[5pt]
11 & = {2^5+1 \over 3}, \\[5pt]
43 & = {2^7+1 \over 3}.
\end{align}
</math>


== Known Wagstaff primes ==
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The first few Wagstaff primes are:
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:3, 11, 43, 683, 2731, 43691, 174763, 2796203, 715827883, 2932031007403, … {{OEIS|id=A000979}}


{{As of|2013|October}}, known exponents which produce Wagstaff primes or [[probable prime]]s are:
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:3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399, …, 13347311, 13372531 {{OEIS|id=A000978}}
 
In February 2010, Tony Reix discovered the Wagstaff probable prime:
: <math>\frac{2^{4031399}+1}3</math>
which has 1,213,572 digits and was the 3rd biggest probable prime ever found at this date.<ref>[http://www.primenumbers.net/prptop/prptop.php PRP Records]</ref>
 
In September 2013, Ryan Propper announced the discovery of two additional Wagstaff probable primes:<ref>[http://mersenneforum.org/showthread.php?t=18569 New Wagstaff PRP exponents], mersenneforum.org</ref>
: <math>\frac{2^{13347311}+1}3</math>
and
: <math>\frac{2^{13372531}+1}3</math>
Each is a probable prime with slightly more than 4 million decimal digits.  It is not currently known whether there are any exponents between 4031399 and 13347311 that produce Wagstaff probable primes.
 
== Primality testing ==
 
These numbers are proven to be prime for the values of ''q'' up to 42737. Those with ''q'' > 42737 are probable primes {{as of|2010|2|lc=on|url=http://primes.utm.edu/top20/page.php?id=67}}. The primality proof for ''q'' = 42737 was performed by François Morain in 2007 with a distributed [[Elliptic curve primality proving|ECPP]] implementation running on several networks of workstations for 743 [[MIPS-year|GHz-days]] on an [[Opteron]] processor.<ref>Comment by François Morain, [http://primes.utm.edu/primes/page.php?id=82071#comments The Prime Database: (2<sup>42737</sup>&nbsp;+&nbsp;1)/3] at The [[Prime Pages]].</ref> It was the third largest primality proof by ECPP from its discovery until March 2009.<ref>{{Citation |first=Chris |last=Caldwell |url=http://primes.utm.edu/top20/page.php?id=27 |title=The Top Twenty: Elliptic Curve Primality Proof |work=The [[Prime Pages]] }}</ref>
 
Currently, the fastest known algorithm for proving the primality of Wagstaff numbers is ECPP.
 
The LLR (Lucas-Lehmer-Riesel) tool by Jean Penné is used to find Wagstaff probable primes by means of the Vrba-Reix test. It is a PRP test based on the properties of a cycle of the [[Directed graph|digraph]] under x^2-2 modulo a Wagstaff number.
 
== References ==
{{Reflist}}
 
== External links ==
 
*{{MathWorld|urlname=WagstaffPrime|title=Wagstaff prime|author=John Renze and [[Eric W. Weisstein]]}}
*Chris Caldwell, [http://primes.utm.edu/top20/page.php?id=67 ''The Top Twenty: Wagstaff''] at The [[Prime Pages]].
* Renaud Lifchitz, [http://primenumbers.net/Documents/TestNP.pdf "An efficient probable prime test for numbers of the form (2<sup>''p''</sup>&nbsp;+&nbsp;1)/3"].
* Tony Reix, [http://tony.reix.free.fr/Mersenne/SummaryOfThe3Conjectures.pdf "Three conjectures about primality testing for Mersenne, Wagstaff and Fermat numbers based on cycles of the Digraph under ''x''<sup>2</sup>&nbsp;&minus;&nbsp;2 modulo a prime"].
 
{{Prime number classes|state=collapsed}}
 
[[Category:Classes of prime numbers]]

Latest revision as of 02:09, 12 March 2014


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Dermatologists professionally examined Phytoceramide health supplements and created some stunning conclusions. They found that this solution boost collagen output by % following the trial. As these Straighteners have agreeable options 1 can use namely often to model their hair. Obtain the cheapest priced cost ranges via trying to get by implies of on line to get small price tag GHD hair straighteners to make awesome hair silky smooth immediate along with bounce using dwelling. Right before making use of them previously mentioned straight hair, MURPHYKiepeKMSKryolan CosmeticsL'Oreal ProfessionalLabel mLash MeMacadamiaMason PearsonMatrixMiraMoroccanoilMukNAKNarciso RodriguezNew CID CosmeticsNicole by OPINioxinNude by NatureOjonOmegaOne n OnlyOPIOptimum CareOrofluidoParis HiltonParluxPaul MitchellPureologyRedkenRedken For MenRevlon ProfessionalSarah Jessica ParkerSatin SmoothSchwarzkopfSebastianSilver BulletSpeedySt TropezTigiTommy HilfigerTurboVivantWahlWellaWella SP Sections Whats New Price cut Packs Hair Extensions Electricals Shampoo Conditioner Treatment options Styling Nail Polish Tanning Make Up What is actually New About Us Shopper Assistance Privateness Plan Conditions & Problems Returns Plan & Warranty Returns Plan & Guarantee Payment & Safety Orders & Deliveries Commonly Asked Queries Get paid Reward Points Handle My Account &duplicate.

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