Theoretical computer science - Revision history

49.14.144.205: /* Formal methods */

2015-01-04T18:24:15Z

Formal methods

← Older revision		Revision as of 20:24, 4 January 2015
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	~~{{unsolved\|mathematics\|Are there any Wall–Sun–Sun primes~~? ~~If yes, is there an infinite number of them~~?}}		Have you ever thought of what constituents go into the construction material? Have you ever wondered how paints and plastics get their desired colors and shapes? Welcome to the world of epoxy resins, the driving force behind different types of industries in the world.<br><br>An Epoxy Resin is also popularly known as polyepoxide and is a thermosetting polymer. This is formed as the resultant chemical reaction in which the polyamine hardener combines with an epoxide resin. The process of formation of Epoxy Resin is termed as curing and largely depends upon factors such as the temperature, choice of resin and hardener compound used in its making. Resins can be solids, liquids, diluents, multifunctional and special resins, cycloaliphatic and Bisphenol types. The content below aims to shed light on the diverse industrial applications of these resins.<br><br>1. It is the basic constituent of paper colorants, plastic dyes and other specialty chemicals. Its chemical property of resistance towards alkalis and thin film adhesions make them perfect to be used as a paint, coat and dyeing agent.<br><br>2. These epoxy resins serves as the basis of number of products like adhesives, fiber-reinforced plastic, dyes, coatings and fiberglass. Its adhesive property is a result of its strong chemical bonds that finds its usage in industries such as the sports, automobile and shipping industries.<br><br>3. They are also used as tooling and in composite applications such as master models, molds, casting, fixtures, laminates, and as other industrial alternates. They not only help to replace the old materials but also help in reducing the costs of production.<br><br>4. They also have a great demand in the electronics industry where it finds its uses in motors, switchgears, bushings, insulators, and transformers. Their durable nature helps to protect the electrical components from any risks such as short circuits, dust and moisture.<br><br>5. These resins are used in the marine industry for components where a great strength and weight ratio is required. If you liked this information and you would certainly like to get additional info relating to [http://raleighx6.blog.com/2014/06/16/la-fibra-de-vidrio-un-componente-util-con-formidable-capacidad-para-el-reciclaje/ resina epoxi] kindly go to the web-site. They are also used widely in the aircraft industry due to its qualities such as strength, filling of gaps and adhesion to certain materials.
	~~In [[number theory]], a '''Wall–Sun–Sun prime''' or '''Fibonacci–Wieferich prime''' is a certain kind~~ of ~~[[prime number]] which is conjectured to exist~~, ~~although none are known.~~

	~~== Definition ==~~
	~~A prime ''p'' > 5 is called a Wall–Sun–Sun prime if ''p''<sup>2</sup> divides~~ the ~~[[Fibonacci number]] <math>F_{p - \left(\frac{{p}}{{5}}\right)}</math>, where~~ the ~~[[Legendre symbol]]~~ <~~math~~>~~\textstyle\left(\frac{{p}}{{5}}\right)~~<~~/math~~> is ~~defined~~ as
	~~:<math>\left(\frac{p}{5}\right) = \begin{cases} 1 &\text{if }p \equiv \pm1 \pmod 5\\ -1 &\text{if }p \equiv \pm2 \pmod 5 \end{cases}</math>~~

	~~Equivalently, a prime ''p''~~ is a ~~Wall–Sun–Sun prime if ''L<sub>p</sub>'' ≡ 1 (mod ''p''<sup>2</sup>), where ''L<sub>p</sub>''~~ is the ~~''p''-th [[Lucas number]]~~.~~<ref>{{cite journal \| last = Andrejić \| first = V~~. ~~\| title = On Fibonacci powers \| journal = Univ~~. ~~Beograd Publ. Elektrotehn. Fak. Ser. Mat. \| volume = 17 \| pages = 38–44 \| year = 2006 \| url = http://www.doiserbia.nb.rs/img/doi/0353-8893/2006/0353-88930617038A~~.~~pdf \| doi = 10.2298/PETF0617038A}}~~<~~/ref~~>~~{{rp\|42}}~~

	~~== Existence ==~~

	~~It has been conjectured that there are infinitely many Wall–Sun–Sun primes.~~<~~ref~~>~~{{Citation \|last=Klaška \|first=Jiří \|year=2007 \|title=Short remark on Fibonacci−Wieferich primes \|journal=Acta Mathematica Universitatis Ostraviensis \|volume=15 \|issue=~~1 ~~\|pages=21–25 \|url=http://dml~~.~~cz/dmlcz/137492 }}.</ref> No Wall–Sun–Sun primes are known {{As~~ of~~\|2013\|12\|lc=on}}.~~

	~~In 2007~~, ~~Richard J~~. ~~McIntosh~~ and ~~Eric L. Roettger showed that if any exist~~, ~~they must be > 2{{e\|14}}.<ref>{{cite journal \|first=R. J. \|last=McIntosh \|first2=E. L. \|last2=Roettger \|title=A search for Fibonacci−Wieferich~~ and Wolstenholme primes \|journal=[[Mathematics of Computation]] \|volume=76 \|issue=260 \|year=2007 \|pages=2087–2094 \|url=http://www.ams.org/journals/mcom/2007-76-260/S0025-5718-07-01955-2/S0025-5718-07-01955-2.~~pdf \|doi=10.1090/S0025-5718-07-01955-2}}~~<~~/ref~~>
	~~Dorais and Klyve extended this range to 9.7{{e\|14}} without finding such a prime.~~<~~ref~~>~~{{Cite paper \|last=Dorais \|first=F~~. ~~G. \|last2=Klyve \|first2=D. W. \|url=http://www~~-~~personal.umich.edu/~dorais/docs/wieferich.pdf \|title=Near Wieferich primes up to 6.7 × 10<sup>15</sup> \|year=2010 }}</ref> In December 2011~~, ~~another search was started by the [[PrimeGrid]] project~~.~~<ref>[http://www.primegrid.com/forum_thread.php?id=3008&nowrap=true#45946 Wall–Sun–Sun Prime Search project] at PrimeGrid</ref> As~~ of ~~December 2013~~, ~~PrimeGrid has extended the search limit to 2.6{{e\|16}}~~ and ~~continues~~.<~~ref~~>~~[http://prpnet.primegrid.com:13001/ Wall-Sun-Sun Prime Search statistics] at PrimeGrid~~<~~/ref~~>

	~~== History ==~~
	~~Wall–Sun–Sun primes~~ are named after [[Donald Dines Wall]],<ref>{{Citation \|first=D. D. \|last=Wall \|title=Fibonacci Series Modulo m \|journal=[[American Mathematical Monthly]] \|volume=67 \|issue=6 \|year=1960 \|pages=525–532 \|doi=10.2307/2309169 }}</ref> [[Zhi Hong Sun]] and [[Zhi Wei Sun]]; Z. H. Sun and ~~Z. W. Sun showed~~ in ~~1992 that if the first case of [[Fermat's last theorem]] was false for a certain prime ''p''~~, ~~then ''p'' would have to be a Wall–Sun–Sun prime.<ref>{{Citation \|first=Zhi-Hong \|last=Sun \|first2=Zhi-Wei \|last2=Sun \|year=1992 \|title=Fibonacci numbers~~ and ~~Fermat’s last theorem \|journal=[[Acta Arithmetica]] \|volume=60 \|issue=4 \|pages=371–388 \|url=http://matwbn.icm.edu.pl/ksiazki/aa/aa60/aa6046~~.~~pdf }}</ref> As a result, prior~~ to ~~[[Andrew Wiles]]' proof of Fermat's last theorem,~~ the ~~search for Wall–Sun–Sun primes was~~ also the ~~search for a potential [[counterexample]] to this centuries-old [[conjecture]]~~.

	~~== Generalizations ==~~
	~~A '''Tribonacci-Wieferich prime''' is a prime ''p'' satisfying {{nowrap\|''h''(''p'') {{=}} ''h''(''p''~~<~~sup~~>2<~~/sup~~>~~)}},~~ where ~~''h'' is the least positive integer satisfying [''T''<sub>''h''</sub>~~,~~''T''<sub>''h''+1</sub>~~,~~''T''<sub>''h''+2</sub>] ≡ [''T''<sub>0</sub>~~, ~~''T''<sub>1</sub>~~, ~~''T''<sub>2</sub>] (mod ''m'')~~ and ~~''T''<sub>''n''</sub> denotes~~ the ~~''n''-th [[Tribonacci number]]~~. ~~No Tribonacci-Wieferich prime exists below 10~~<~~sup~~>11<~~/sup~~>.~~<ref>{{cite journal \| last = Klaška \| first = Jiří \| title = A search~~ for ~~Tribonacci-Wieferich primes \| journal = Acta Mathematica Universitatis Ostraviensis \| volume = 16 \| issue = 1 \| pages = 15–20 \| year = 2008 \| url =~~ http://~~dml~~.cz/~~dmlcz~~/~~137497}}<~~/~~ref>~~

	~~== See also ==~~
	* [[Wieferich prime]]
	* [[Wilson prime]]
	* [[Wolstenholme prime]]
	* [[PrimeGrid]]
	* [[Table of congruences]]

	~~== References ==~~
	~~{{Reflist}}~~

	~~== Further reading ==~~
	*{{Cite book \|title=Prime Numbers: A Computational Perspective \|first=Richard E. \|last=Crandall \|first2=Carl \|last2=Pomerance \|publisher=Springer \|year=2001 \|page=29 \|isbn=0-~~387~~-~~94777~~-~~9 }}~~
	*{{cite paper \|last=Saha \|first=Arpan \|last2=Karthik \|first2=C. S. \|title=A Few Equivalences of Wall-~~Sun~~-~~Sun Prime Conjecture \|year=2011 \|work=Working paper \|arxiv=1102.1636 }}~~

	~~== External links ==~~
	* Chris Caldwell, [http://primes.utm.edu/~~glossary/page.php?sort=WallSunSunPrime The Prime Glossary: Wall–Sun–Sun prime~~] at the ~~[[Prime Pages]].~~
	* {{MathWorld\|urlname=Wall-~~Sun-SunPrime\|title=Wall–Sun–Sun prime}}~~
	* Richard McIntosh, [http://www.~~loria.fr/~zimmerma/records/Wieferich.status Status of~~ the ~~search for Wall–Sun–Sun primes (October 2003)]~~

	~~{{Prime number classes}}~~

	~~{{DEFAULTSORT:Wall-Sun-Sun prime}}~~
	~~[[Category:Classes~~ of ~~prime numbers]]~~
	~~[[Category:Unsolved problems in mathematics]]~~

115.112.74.130: /* External links */

2013-12-30T11:32:54Z

External links

← Older revision		Revision as of 13:32, 30 December 2013
Line 1:		Line 1:
			{{unsolved\|mathematics\|Are there any Wall–Sun–Sun primes? If yes, is there an infinite number of them?}}
			In [[number theory]], a '''Wall–Sun–Sun prime''' or '''Fibonacci–Wieferich prime''' is a certain kind of [[prime number]] which is conjectured to exist, although none are known.

			== Definition ==
			A prime ''p'' > 5 is called a Wall–Sun–Sun prime if ''p''<sup>2</sup> divides the [[Fibonacci number]] <math>F_{p - \left(\frac{{p}}{{5}}\right)}</math>, where the [[Legendre symbol]] <math>\textstyle\left(\frac{{p}}{{5}}\right)</math> is defined as
			:<math>\left(\frac{p}{5}\right) = \begin{cases} 1 &\text{if }p \equiv \pm1 \pmod 5\\ -1 &\text{if }p \equiv \pm2 \pmod 5 \end{cases}</math>

	~~A new Tribe~~ is ~~the greatest strong of all and you will probably have the planet~~ (~~virtual~~) ~~at your toes~~, ~~and therefore all that with sole a brief on~~-~~line vid~~ that ~~may direct you step by step in how to get regarding cheat code for Mismatch of Tribes~~.<br><br>~~When you~~ are ~~locating the latest handle system tough as a way to use~~, [http://www.~~google~~.de/~~search?q~~=~~optimize optimize] the ways within your activity~~. ~~The default manage course of action might not be on everyone~~. ~~Some sufferers prefer~~ a ~~better let you know screen, a set of more sensitive management or perhaps an inverted format~~. In ~~several video recordings gaming~~, ~~you may control these from~~ the ~~setting's area~~.<br><br>~~Okazaki~~, ~~japan tartan draws getting ideas through your country's involvement in cherry blossom~~ and ~~will involve pink, white, green as well brown lightly colours~~. ~~clash of clans cheats~~. ~~Features is called Sakura, japan for cherry blossom~~.<br><br>~~Invest attention to how a lot of money your teenager is spending on video console games~~. ~~These products commonly cheap~~ and ~~there is generally often the option most typically associated with buying more add-ons from the game itself~~. ~~Set monthly~~ and ~~on an annual basis limits on~~ the ~~sum~~ of ~~money that can be spent on video games. Also~~, ~~may~~ have ~~conversations with your boys about budgeting~~.<br><br>Just variety of participants what individuals perform Clash of Clans across the world this can help you the chance that can crew up with clans that have been caused by players from different nations around the world and ~~can also keep living competitive towards other clans. If you enjoyed this information and you would such as to receive additional info regarding clash of clans hack no survey (~~[http://~~prometeu~~.~~net you can try here~~]~~) kindly see our internet site. This will earn~~ the ~~game considerably more interesting as you will look~~ for a ~~great deal of multiple strategies that might supply by participants and particular boosts the unpredictability section~~. ~~Getting the right strategy november 23~~ is where the ~~gamblers skills are tested~~, ~~though the game is simple perform and understand.~~<br><br>It'~~s tough to select the appropriate xbox game gaming product. In the beginning~~, ~~you should regarding your standard requirements like a video game player~~, ~~as well as check out~~ the ~~additional features made available from each unit you are considering~~. ~~Consider investigating on~~-~~line. Check many testimonials to ascertain if former gamers have discovered problems with the unit. Prior to buying a game process, you should know up to a whopping you are able to help regarding it~~.<br><br>~~Disclaimer~~: ~~I aggregate the tips about this commodity by business a lot~~ of ~~CoC and accomplishing some inquiry~~. ~~To the best involving my knowledge~~, ~~is it authentic inside addition to I accept amateur arrested all abstracts and methods~~. ~~Nevertheless, it is consistently accessible which i accept fabricated a aberration about or which~~ the ~~bold has afflicted butt publication~~. ~~Use by your very own risk~~, ~~I am accommodate virtually any guarantees~~. ~~Please get~~ in ~~blow if the public acquisition annihilation amiss.~~		Equivalently, a prime ''p'' is a Wall–Sun–Sun prime if ''L<sub>p</sub>'' ≡ 1 (mod ''p''<sup>2</sup>), where ''L<sub>p</sub>'' is the ''p''-th [[Lucas number]].<ref>{{cite journal \| last = Andrejić \| first = V. \| title = On Fibonacci powers \| journal = Univ. Beograd Publ. Elektrotehn. Fak. Ser. Mat. \| volume = 17 \| pages = 38–44 \| year = 2006 \| url = http://www.doiserbia.nb.rs/img/doi/0353-8893/2006/0353-88930617038A.pdf \| doi = 10.2298/PETF0617038A}}</ref>{{rp\|42}}

			== Existence ==

			It has been conjectured that there are infinitely many Wall–Sun–Sun primes.<ref>{{Citation \|last=Klaška \|first=Jiří \|year=2007 \|title=Short remark on Fibonacci−Wieferich primes \|journal=Acta Mathematica Universitatis Ostraviensis \|volume=15 \|issue=1 \|pages=21–25 \|url=http://dml.cz/dmlcz/137492 }}.</ref> No Wall–Sun–Sun primes are known {{As of\|2013\|12\|lc=on}}.

			In 2007, Richard J. McIntosh and Eric L. Roettger showed that if any exist, they must be > 2{{e\|14}}.<ref>{{cite journal \|first=R. J. \|last=McIntosh \|first2=E. L. \|last2=Roettger \|title=A search for Fibonacci−Wieferich and Wolstenholme primes \|journal=[[Mathematics of Computation]] \|volume=76 \|issue=260 \|year=2007 \|pages=2087–2094 \|url=http://www.ams.org/journals/mcom/2007-76-260/S0025-5718-07-01955-2/S0025-5718-07-01955-2.pdf \|doi=10.1090/S0025-5718-07-01955-2}}</ref>
			Dorais and Klyve extended this range to 9.7{{e\|14}} without finding such a prime.<ref>{{Cite paper \|last=Dorais \|first=F. G. \|last2=Klyve \|first2=D. W. \|url=http://www-personal.umich.edu/~dorais/docs/wieferich.pdf \|title=Near Wieferich primes up to 6.7 × 10<sup>15</sup> \|year=2010 }}</ref> In December 2011, another search was started by the [[PrimeGrid]] project.<ref>[http://www.primegrid.com/forum_thread.php?id=3008&nowrap=true#45946 Wall–Sun–Sun Prime Search project] at PrimeGrid</ref> As of December 2013, PrimeGrid has extended the search limit to 2.6{{e\|16}} and continues.<ref>[http://prpnet.primegrid.com:13001/ Wall-Sun-Sun Prime Search statistics] at PrimeGrid</ref>

			== History ==
			Wall–Sun–Sun primes are named after [[Donald Dines Wall]],<ref>{{Citation \|first=D. D. \|last=Wall \|title=Fibonacci Series Modulo m \|journal=[[American Mathematical Monthly]] \|volume=67 \|issue=6 \|year=1960 \|pages=525–532 \|doi=10.2307/2309169 }}</ref> [[Zhi Hong Sun]] and [[Zhi Wei Sun]]; Z. H. Sun and Z. W. Sun showed in 1992 that if the first case of [[Fermat's last theorem]] was false for a certain prime ''p'', then ''p'' would have to be a Wall–Sun–Sun prime.<ref>{{Citation \|first=Zhi-Hong \|last=Sun \|first2=Zhi-Wei \|last2=Sun \|year=1992 \|title=Fibonacci numbers and Fermat’s last theorem \|journal=[[Acta Arithmetica]] \|volume=60 \|issue=4 \|pages=371–388 \|url=http://matwbn.icm.edu.pl/ksiazki/aa/aa60/aa6046.pdf }}</ref> As a result, prior to [[Andrew Wiles]]' proof of Fermat's last theorem, the search for Wall–Sun–Sun primes was also the search for a potential [[counterexample]] to this centuries-old [[conjecture]].

			== Generalizations ==
			A '''Tribonacci-Wieferich prime''' is a prime ''p'' satisfying {{nowrap\|''h''(''p'') {{=}} ''h''(''p''<sup>2</sup>)}}, where ''h'' is the least positive integer satisfying [''T''<sub>''h''</sub>,''T''<sub>''h''+1</sub>,''T''<sub>''h''+2</sub>] ≡ [''T''<sub>0</sub>, ''T''<sub>1</sub>, ''T''<sub>2</sub>] (mod ''m'') and ''T''<sub>''n''</sub> denotes the ''n''-th [[Tribonacci number]]. No Tribonacci-Wieferich prime exists below 10<sup>11</sup>.<ref>{{cite journal \| last = Klaška \| first = Jiří \| title = A search for Tribonacci-Wieferich primes \| journal = Acta Mathematica Universitatis Ostraviensis \| volume = 16 \| issue = 1 \| pages = 15–20 \| year = 2008 \| url = http://dml.cz/dmlcz/137497}}</ref>

			== See also ==
			* [[Wieferich prime]]
			* [[Wilson prime]]
			* [[Wolstenholme prime]]
			* [[PrimeGrid]]
			* [[Table of congruences]]

			== References ==
			{{Reflist}}

			== Further reading ==
			*{{Cite book \|title=Prime Numbers: A Computational Perspective \|first=Richard E. \|last=Crandall \|first2=Carl \|last2=Pomerance \|publisher=Springer \|year=2001 \|page=29 \|isbn=0-387-94777-9 }}
			*{{cite paper \|last=Saha \|first=Arpan \|last2=Karthik \|first2=C. S. \|title=A Few Equivalences of Wall-Sun-Sun Prime Conjecture \|year=2011 \|work=Working paper \|arxiv=1102.1636 }}

			== External links ==
			* Chris Caldwell, [http://primes.utm.edu/glossary/page.php?sort=WallSunSunPrime The Prime Glossary: Wall–Sun–Sun prime] at the [[Prime Pages]].
			* {{MathWorld\|urlname=Wall-Sun-SunPrime\|title=Wall–Sun–Sun prime}}
			* Richard McIntosh, [http://www.loria.fr/~zimmerma/records/Wieferich.status Status of the search for Wall–Sun–Sun primes (October 2003)]

			{{Prime number classes}}

			{{DEFAULTSORT:Wall-Sun-Sun prime}}
			[[Category:Classes of prime numbers]]
			[[Category:Unsolved problems in mathematics]]

en>Computhematics at 14:44, 23 August 2012

2012-08-23T14:44:36Z

New page

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