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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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The usage of ecigs is actually a exercise that's becoming increasingly popular, and many folks took up the use of e-cigs like a solution alternative to smoking. The reason being ecigs don't end in ash, and they do not result in a person’s property, clothes, and automobile stinking poor. One of the greatest parts of needs to use e cigs involves investing in a starter kit, and there are a quantity of issues a person should know about investing in a starter kit. This includes investing in a slightly greater beginner kit, and there are a few things someone must have included in their starter kit.
Buying a slightly greater kit
One of the most typical blunders that folks produce when first beginning to smoke e-cigs is the fact that they have a tendency to purchase the cheapest e-cig that's offered to them. This could cause a individual not necessarily taking up the practice, and it also means that someone WOn't ever purchase a great ecig once they do take-up the practice. That is why it's advisable to get a better one, that appears the way an individual are interested to, which is a good good concept to get one that may be slowly enhanced over time.
What must certanly be included in the kit
A superb starter kit may have things such as a charger that is user friendly. The kit also needs to include guidelines on how best to complete the e-cigarette with juice, a rag to clean it with, and it also needs to have a some extra e-juice or capsules. More on our website Read Full Report.