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| The '''Cunningham project''' is a project, started in 1925, to [[integer factorization|factor]] numbers of the form ''b''<sup>''n''</sup> ± 1 for ''b'' = 2, 3, 5, 6, 7, 10, 11, 12 and large ''n''. The project is named after [[Allan Joseph Champneys Cunningham]], who published the first version of the table together with [[H. J. Woodall|Herbert J. Woodall]].<ref>{{cite book|last=Cunningham|first=Allan J. C.|last2=Woodall|first2=H. J.|title=Factorisation of y<sup>n</sup> ± 1, y = 2, 3, 5, 6, 7, 10, 11, 12, up to high powers n|publisher=Hodgson|year=1925}}</ref> There are three printed versions of the table, the most recent published in 2002,<ref>{{cite web|url=http://www.ams.org/online_bks/conm22|title=Factorizations of b<sup>n</sup> ± 1, b = 2, 3, 5, 6, 7, 10, 11, 12 up to high powers|last1=Brillhart|first1=John|authorlink1=John Brillhart|last2=Lehmer|first2=Derrick H.|authorlink2=Derrick Henry Lehmer|last3=Selfridge|first3=John L.|authorlink3=John Selfridge|last4=Tuckerman|first4=Bryant|last5=Wagstaff|first5=Samuel S.|authorlink5=Samuel S. Wagstaff Jr.|publisher=AMS|year=2002}}</ref> as well as an online version.<ref>{{cite web|url=http://www.cerias.purdue.edu/homes/ssw/cun/index.html|title=The Cunningham Project|accessdate=18 March 2012}}</ref>
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| The current limits of the exponents are:
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| {| class="wikitable" style="text-align:center"
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| !Base
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| !2
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| !3
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| !5
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| !6
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| !7
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| !10
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| !11
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| !12
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| |-
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| !Limit
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| |1200
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| |800
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| |500
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| |450
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| |400
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| |400
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| |350
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| |300
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| |-
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| ![[Aurifeuillian factorization|Aurifeuillian]] limit
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| |2400
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| |1600
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| |1000
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| |900
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| |800
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| |800
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| |700
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| |600
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| |}
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| ==Factors of Cunningham numbers==
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| Two types of factors can be derived from a Cunningham number without having to use a factorisation algorithm: algebraic factors, which depend on the exponent, and Aurifeuillian factors, which depend on both the base and the exponent.
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| ===Algebraic factors===
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| From elementary algebra,
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| :<math>(b^{kn}-1) = (b^n-1) \sum _{r=0}^{k-1} b^{rn}</math>
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| for all ''k'', and
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| :<math>(b^{kn}+1) = (b^n+1) \sum _{r=0}^{k-1} (-1)^r \cdot b^{rn}</math>
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| for odd ''k''. In addition, ''b''<sup>2''n''</sup> − 1 = (''b''<sup>''n''</sup> − 1)(''b''<sup>''n''</sup> + 1). Thus, when ''m'' divides ''n'', ''b''<sup>''m''</sup> − 1 and ''b''<sup>''m''</sup> + 1 are factors of ''b''<sup>''n''</sup> − 1 if the quotient of ''n'' over ''m'' is even; only the first number is a factor if the quotient is odd. ''b''<sup>''m''</sup> + 1 is a factor of ''b''<sup>''n''</sup> − 1, if ''m'' divides ''n'' and the quotient is odd.
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| ===Aurifeuillian factors===
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| {{main|Aurifeuillian factorization}}
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| When the number is of a particular form (the exact expression varies with the base), Aurifeuillian factorization may be used, which gives a product of two or three numbers. If ''h'' = 2''k'' − 1, the following equations give Aurifeuillian factors for the Cunningham project bases as a product of ''F'', ''L'' and ''M'':<ref>{{cite web|title=Main Cunningham Tables|url=http://homes.cerias.purdue.edu/~ssw/cun/pmain1211|accessdate=18 March 2012}} At the end of tables 2LM, 3+, 5−, 7+, 10+, 11+ and 12+ are formulae detailing the Aurifeuillian factorisations.</ref>
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| {| class="wikitable" style="text-align:center"
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| !Base
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| !Number
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| !F
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| !L
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| !M
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| !Other definitions
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| |-
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| !2
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| |2<sup>2''h''</sup> + 1
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| |1
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| |2<sup>''h''</sup> − 2<sup>''k''</sup> + 1
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| |2<sup>''h''</sup> + 2<sup>''k''</sup> + 1
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| |
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| |-
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| !3
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| |3<sup>3''h''</sup> + 1
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| |3<sup>''h''</sup> + 1
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| |3<sup>''h''</sup> − 3<sup>''k''</sup> + 1
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| |3<sup>''h''</sup> + 3<sup>''k''</sup> + 1
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| |-
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| !5
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| |5<sup>5''h''</sup> − 1
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| |5<sup>''h''</sup> − 1
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| |''T''<sup>2</sup> − 5<sup>''k''</sup>''T'' + 5<sup>''h''</sup>
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| |''T''<sup>2</sup> + 5<sup>''k''</sup>''T'' + 5<sup>''h''</sup>
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| |''T'' = 5<sup>''h''</sup> + 1
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| |-
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| !6
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| |6<sup>6''h''</sup> + 1
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| |6<sup>2''h''</sup> + 1
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| |''T''<sup>2</sup> − 6<sup>''k''</sup>''T'' + 6<sup>''h''</sup>
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| |''T''<sup>2</sup> + 6<sup>''k''</sup>''T'' + 6<sup>''h''</sup>
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| |''T'' = 6<sup>''h''</sup> + 1
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| |-
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| !7
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| |7<sup>7''h''</sup> + 1
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| |7<sup>''h''</sup> + 1
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| |''T''<sup>3</sup> − ''B''
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| |''T''<sup>3</sup> + ''B''
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| |''T'' = 7<sup>''h''</sup> + 1<br/>''B'' = 7<sup>''k''</sup>(''T''<sup>2</sup> − 7<sup>''h''</sup>)
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| |-
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| !10
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| |10<sup>10''h''</sup> + 1
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| |10<sup>2''h''</sup> + 1
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| |''A'' − ''B''
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| |''A'' + ''B''
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| |''A'' = 10<sup>4''h''</sup> + 5(10<sup>3''h''</sup>) + 7(10<sup>2''h''</sup>) + 5(10<sup>''h''</sup>) + 1<br/>''B'' = 10<sup>''k''</sup>(10<sup>3''h''</sup> + 2(10<sup>2''h''</sup>) + 2(10<sup>''h''</sup>) + 1)
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| |-
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| !11
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| |11<sup>11''h''</sup> + 1
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| |11<sup>''h''</sup> + 1
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| |''A'' − ''B''
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| |''A'' + ''B''
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| |''A'' = 11<sup>5''h''</sup> + 5(11<sup>4''h''</sup>) − 11<sup>3''h''</sup> − 11<sup>2''h''</sup> + 5(11<sup>''h''</sup>) + 1<br/>''B'' = 11<sup>''k''</sup>(11<sup>4''h''</sup> + 11<sup>3''h''</sup> − 11<sup>2''h''</sup> + 11<sup>''h''</sup> + 1)
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| |-
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| !12
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| |12<sup>3''h''</sup> + 1
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| |12<sup>''h''</sup> + 1
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| |12<sup>''h''</sup> − 2<sup>''h''</sup>3<sup>''k''</sup> + 1
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| |12<sup>''h''</sup> + 2<sup>''h''</sup>3<sup>''k''</sup> + 1
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| |}
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| ===Other factors===
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| Once the algebraic and Aurifeuillian factors are removed, the other factors of ''b''<sup>''n''</sup> ± 1 are always of the form 2''kn'' + 1. When ''n'' is prime, both algebraic and Aurifeuillian factors are not possible, except the trivial factors (''b'' − 1 for ''b''<sup>''n''</sup> − 1 and ''b'' + 1 for ''b''<sup>''n''</sup> + 1). For [[Mersenne numbers]], the trivial factors are not possible for prime ''n'', so all factors are of the form 2''kn'' + 1. In general, all factors of (''b''<sup>''n''</sup> − 1)/(''b'' − 1) are of the form 2''kn'' + 1, where ''b'' ≥ 2 and ''n'' is prime, except when ''n'' divides ''b'' − 1, in which case (''b''<sup>''n''</sup> − 1)/(''b'' − 1) is divisible by ''n'' itself.
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| Cunningham numbers of the form ''b''<sup>''n''</sup> − 1 can only be prime if ''b'' = 2 and ''n'' is prime, assuming that ''n'' ≥ 2; these are the Mersenne numbers. Numbers of the form ''b''<sup>''n''</sup> + 1 can only be prime if ''b'' is even and ''n'' is a power of 2, again assuming ''n'' ≥ 2; these are the generalized Fermat numbers, which are [[Fermat number]]s when ''a'' = 1. Any factor of a Fermat number 2<sup>2<sup>''k''</sup></sup> + 1 is of the form ''k''2<sup>''n'' + 2</sup> + 1.
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| ==Notation==
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| ''b''<sup>''n''</sup> − 1 is denoted as ''b'',''n''−. Similarly, ''b''<sup>''n''</sup> + 1 is denoted as ''b'',''n''+. When dealing with numbers of the form required for Aurifeuillian factorisation, ''b'',''n''L and ''b'',''n''M are used to denote L and M in [[Cunningham project#Aurifeuillian factors|the products above]].<ref>{{cite web|url=http://homes.cerias.purdue.edu/~ssw/cun/notat|title=Explanation of the notation on the Pages|accessdate=18 March 2012}}</ref> References to ''b'',''n''− and ''b'',''n''+ are to the number with all algebraic and Aurifeuillian factors removed. For example, Mersenne numbers are of the form 2,''n''− and Fermat numbers are of the form 2,2<sup>''n''</sup>+; the number Aurifeuille factored in 1871 was the product of 2,58L and 2,58M.
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| ==See also== | |
| *[[Cunningham number]]
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| *[[Lenstra elliptic curve factorization#External links|ECMNET]] and [[NFSNET|NFS@Home]], two collaborations working for the Cunningham project
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| ==References==
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| {{reflist}}
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| ==External links==
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| *[http://www.cerias.purdue.edu/homes/ssw/cun/index.html Cunningham project homepage]
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| *[http://wwwmaths.anu.edu.au/~brent/factors.html Brent-Montgomery-te Riele table] (Cunningham tables for higher bases)
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| *[http://mersennewiki.org/index.php/Cunningham_Tables Cunningham tables on Mersennewiki]
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| [[Category:Number theory]]
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