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| In [[number theory]], a '''Sierpinski''' or '''Sierpiński number''' is an odd [[natural number]] ''k'' such that ''k''2<sup>''n''</sup> + 1 is [[composite number|composite]], for all natural numbers ''n''; in 1960, [[Wacław Sierpiński]] proved that there are [[Infinity|infinitely]] many odd [[integer]]s ''k'' which have this property.
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| In other words, when ''k'' is a Sierpiński number, all members of the following [[Set (mathematics)|set]] are composite:
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| :<math>\left\{\,k 2^n + 1 : n \in\mathbb{N}\,\right\}.</math>
| | Introduction. The [http://www.bing.com/search?q=author%27s&form=MSNNWS&mkt=en-us&pq=author%27s author's] name is Dalton but it's not the maximum masucline name out that there. The favorite hobby for him and his kids is to drive and he's been really doing it for quite some time. He operates as a cashier. His wife and him chose to live on in South Carolina additionally his family loves them. Go to his website to find out more: http://prometeu.net<br><br>Also visit my blog post :: [http://prometeu.net clash of clans triche] |
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| Numbers in such a set with odd ''k'' and ''k'' < 2<sup>''n''</sup> are [[Proth number]]s.
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| ==Known Sierpiński numbers==
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| The sequence of currently ''known'' Sierpiński numbers begins with:
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| : 78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, 965431, … {{OEIS|A076336}}.
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| The number 78557 was proved to be a Sierpiński number by [[John Selfridge]] in 1962, who showed that all numbers of the form 78557·2<sup>''n''</sup>+1 have a [[factorization|factor]] in the [[covering set]] {3, 5, 7, 13, 19, 37, 73}. For another known Sierpiński number, 271129, the covering set is {3, 5, 7, 13, 17, 241}. All currently known Sierpiński numbers possess similar covering sets.<ref name="PG">[http://primes.utm.edu/glossary/page.php?sort=SierpinskiNumber Sierpinski number at The Prime Glossary]</ref> | |
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| ==The Sierpiński problem== | |
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| {{Further|Seventeen or Bust}}
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| {{unsolved|mathematics|Is 78,557 the smallest Sierpiński number?}}
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| The '''Sierpiński problem''' is: "What is the smallest Sierpiński number?"
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| In 1967, Sierpiński and Selfridge [[conjecture]]d that 78,557 is the smallest Sierpiński number, and thus the answer to the Sierpiński problem.
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| To show that 78,557 really is the smallest Sierpiński number, one must show that all the odd numbers smaller than 78,557 are ''not'' Sierpiński numbers. That is, for every odd ''k'' below 78,557 there exists a positive integer ''n'' such that ''k''2<sup>''n''</sup>+1 is prime.<ref name="PG" /> {{As of|2013|12}}, there are only six candidates:
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| : ''k'' = 10223, 21181, 22699, 24737, 55459, and 67607
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| which have not been eliminated as possible Sierpiński numbers.<ref>[http://seventeenorbust.com/stats/rangeStatsEx.mhtml Seventeen or Bust: Project Stats]</ref> [[Seventeen or Bust]] (with [[PrimeGrid]]), a [[distributed computing]] project, is testing these remaining numbers. If the project finds a prime of the form ''k''2<sup>''n''</sup>+1 for every remaining ''k'', the Sierpiński problem will be solved.
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| ==See also==
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| {{Portal|Mathematics}}
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| * [[Riesel number]]
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| ==References==
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| {{Reflist}}
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| ==Further reading==
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| *{{citation |first=Richard K. |last=Guy |authorlink=Richard K. Guy |title=Unsolved Problems in Number Theory |publisher=[[Springer-Verlag]] |location=New York |year=2004 |page=120 |isbn=0-387-20860-7 }}
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| ==External links==
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| *[http://www.prothsearch.net/sierp.html The Sierpinski problem: definition and status]
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| *{{MathWorld| urlname=SierpinskisCompositeNumberTheorem |title=Sierpinski's composite number theorem}}
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| *[http://www.mersenneforum.org/showthread.php?t=2665 The Prime Sierpinski Problem], a related question.
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| {{Classes of natural numbers}}
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| {{DEFAULTSORT:Sierpinski Number}}
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| [[Category:Prime numbers]]
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| [[Category:Number theory]]
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| [[Category:Conjectures|Sierpinski-Selfridge conjecture]]
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| [[Category:Unsolved problems in mathematics|Sierpinski problem]]
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| [[Category:Science and technology in Poland]]
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Introduction. The author's name is Dalton but it's not the maximum masucline name out that there. The favorite hobby for him and his kids is to drive and he's been really doing it for quite some time. He operates as a cashier. His wife and him chose to live on in South Carolina additionally his family loves them. Go to his website to find out more: http://prometeu.net
Also visit my blog post :: clash of clans triche