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| In [[mathematics]], a '''Riesel number''' is an [[Even and odd numbers|odd]] [[natural number]] ''k'' for which the integers of the form ''k''·2<sup>''n''</sup> − 1 are [[composite number|composite]] for all natural numbers ''n'' {{OEIS|A076337}}.
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| In other words, when ''k'' is a Riesel number, all members of the following set are composite:
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| :<math>\left\{\,k 2^n - 1 : n \in\mathbb{N}\,\right\}.</math>
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| In 1956, [[Hans Riesel]] showed that there are an [[Infinite set|infinite]] number of integers ''k'' such that ''k''·2<sup>''n''</sup> − 1 is not [[prime number|prime]] for any integer ''n''. He showed that the number 509203 has this property, as does 509203 plus any positive [[integer]] multiple of 11184810.<ref>{{cite journal | first=Hans | last=Riesel | title=Några stora primtal | journal=Elementa | year=1956 |volume=39 | pages=258–260 | authorlink=Hans Riesel }}</ref>
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| A number can be shown to be a Riesel number by exhibiting a ''[[covering set]]'': a set of prime numbers that will divide any member of the sequence, so called because it is said to "cover" that sequence. The only proven Riesel numbers below one million have covering sets as follows:
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| * 509203×2<sup>''n''</sup> − 1 has covering set {3, 5, 7, 13, 17, 241}
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| * 762701×2<sup>''n''</sup> − 1 has covering set {3, 5, 7, 13, 17, 241}
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| * 777149×2<sup>''n''</sup> − 1 has covering set {3, 5, 7, 13, 19, 37, 73}
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| * 790841×2<sup>''n''</sup> − 1 has covering set {3, 5, 7, 13, 19, 37, 73}
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| * 992077×2<sup>''n''</sup> − 1 has covering set {3, 5, 7, 13, 17, 241}.
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| The '''Riesel problem''' consists in determining the smallest Riesel number. Because no covering set has been found for any ''k'' less than 509203, it is conjectured that 509203 is the smallest Riesel number. However, 52 values of ''k'' less than this have yielded only composite numbers for all values of ''n'' so far tested. The smallest of these are 2293, 9221, 23669, 31859, 38473, 46663, 67117, 74699, 81041 and 93839.
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| Thirty-three numbers have had primes found by the [[Riesel Sieve|Riesel Sieve project]] (analogous to [[Seventeen or Bust]] for [[Sierpinski number]]s). Currently [[PrimeGrid]] is working on remaining numbers and has found twelve primes {{as of|2013|12|25|lc=y}}.<ref>[http://www.primegrid.com/download/TRP-40597.pdf Twelveth Riesel prime discovery announcement on PrimeGrid]</ref>
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| ==Simultaneously Riesel and Sierpiński==
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| A number may be simultaneously Riesel and [[Sierpiński number|Sierpiński]]. These are called Brier Numbers. The smallest known example (as of March 2013) is 143665583045350793098657 ({{OEIS link|A076335}}).<ref>[http://www.primepuzzles.net/problems/prob_029.htm Problem 29.- Brier Numbers]</ref>
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| ==See also==
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| {{Portal|Mathematics}}
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| *[[Experimental mathematics]]
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| *[[BOINC]]
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| *[[PrimeGrid]]
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| ==References==
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| {{reflist}}
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| *{{cite book |title=Unsolved Problems in Number Theory |last=Guy |first=Richard K. |authorlink=Richard K. Guy |coauthors= |year=2004 |publisher=[[Springer-Verlag]] |location=Berlin |isbn=0-387-20860-7 |page=120 |pages= |url= }}
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| *{{cite book | title=The New Book of Prime Number Records | first=Paulo | last=Ribenboim | publisher=[[Springer-Verlag]] | location=New York | isbn=0-387-94457-5 | year=1996 | pages=357–358 | authorlink=Paulo Ribenboim }}
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| ==External links==
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| * [http://www.primegrid.com PrimeGrid]
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| * [http://www.prothsearch.net/rieselprob.html The Riesel Problem: Definition and Status]
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| {{Classes of natural numbers}}
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| {{DEFAULTSORT:Riesel Number}}
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| [[Category:Analytic number theory]]
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| [[Category:Unsolved problems in mathematics]]
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| [[Category:Prime numbers]]
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| {{numtheory-stub}}
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Luke is often a superstar inside the building along with the vocation growth initial 2nd to his third stadium record, And , is the confirmation. He burst open to the picture in 2007 along with his special combination of straight down-home convenience, motion picture superstar good appears and words, is scheduled t in justin bieber concerts the main way. The new record Top on the country graph and #2 in the take maps, generating it the next highest first appearance in those days of 2008 for the region musician.
The boy of any , is aware of determination and perseverance are key elements in relation to a prosperous career- . His first album, Keep Me, generated the most notable strikes “All My Buddies Say” and “Country Man,” although his hard work, Doin’ Thing, found the singer-3 straight No. 2 single people: Different Contacting Can be a Fantastic Thing.”
While in the fall of 2011, Concert tour: Bryan & which had an luke bryan in concert impressive selection of , which includes Urban. “It’s almost like you’re acquiring a endorsement to look to another level, states individuals artists which were a part of the Tourmore than into a larger measure of designers.” It twisted as among the most successful tours in its ten-calendar year historical past.
My web page: ticket brokers; lukebryantickets.asiapak.net,