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| In [[mathematics]], specifically in [[number theory]], a '''Cunningham number''' is a certain kind of integer named after English mathematician [[Allan Joseph Champneys Cunningham|A. J. C. Cunningham]].
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| ==Definition==
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| Cunningham numbers are a simple type of [[Binomial theorem#Binomial Number|binomial number]], they are of the form
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| :<math>b^n\pm1\,</math>
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| where ''b'' and ''n'' are integers and ''b'' is not already a power of some other number. They are denoted ''C''<sup>±</sup>(''b'', ''n'').
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| ==Primality==
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| Establishing whether or not a given Cunningham number is prime has been the main focus of research around this type of number.<ref>J. Brillhart, D. H. Lehmer, J. Selfridge, B. Tuckerman,and S. S. Wagstaff Jr., ''Factorizations of b<sup>n</sup>±1, b=2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers (n)'', 3rd ed. Providence, RI: Amer. Math. Soc., 1988.</ref> Two particularly famous families of Cunningham numbers in this respect are the [[Fermat numbers]], which are those of the form ''C''<sup>+</sup>(2,2<sup>''m''</sup>), and the [[Mersenne numbers]], which are of the form ''C''<sup>-</sup>(2,''n'').
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| Cunningham worked on gathering together all known data on which of these numbers were prime. In 1925 he published tables which summarised his findings with [[H. J. Woodall]], and much computation has been done in the intervening time to fill these tables.<ref>R. P. Brent and H. J. J. te Riele, ''Factorizations of a<sup>n</sup>±1, 13≤a<100'' Report NM-R9212, Centrum voor Wiskunde en Informatica. Amsterdam, 1992.</ref>
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| ==See also==
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| *[[Cunningham project]]
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| ==References==
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| {{Reflist}}
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| ==External links==
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| *[http://mathworld.wolfram.com/CunninghamNumber.html Cunningham Number at MathWorld]
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| *[http://homes.cerias.purdue.edu/~ssw/cun/index.html The Cunningham Project, a collaborative effort to factor Cunningham numbers]
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| [[Category:Number theory]]
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