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| In [[mathematics]], a '''Cunningham chain''' is a certain sequence of [[prime number]]s. Cunningham chains are named after [[mathematician]] [[Allan Joseph Champneys Cunningham|A. J. C. Cunningham]]. They are also called '''chains of nearly doubled primes'''.
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| A '''Cunningham chain of the first kind''' of length ''n'' is a sequence of prime numbers (''p''<sub>1</sub>, ..., ''p''<sub>''n''</sub>) such that for all 1 ≤ ''i'' < ''n'', ''p''<sub>''i''+1</sub> = 2''p''<sub>''i''</sub> + 1. (Hence each term of such a chain except the last one is a [[Sophie Germain prime]], and each term except the first is a [[safe prime]]).
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| It follows that
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| : <math> | |
| \begin{align}
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| p_2 & = 2p_1+1, \\
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| p_3 & = 4p_1+3, \\
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| p_4 & = 8p_1+7, \\
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| & {}\ \vdots \\
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| p_i & = 2^{i-1}p_1 + (2^{i-1}-1).
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| \end{align}
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| </math>
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| Or, by setting <math> a = \frac{p_1 + 1}{2} </math> (the number <math> a </math> is not part of the sequence and need not be a prime number), we have <math> p_i = 2^{i} a - 1 </math>
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| Similarly, a '''Cunningham chain of the second kind''' of length ''n'' is a sequence of prime numbers (''p''<sub>1</sub>,...,''p''<sub>''n''</sub>) such that for all 1 ≤ ''i'' < ''n'', ''p''<sub>''i''+1</sub> = 2''p''<sub>''i''</sub> − 1.
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| It follows that the general term is
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| : <math> p_i = 2^{i-1}p_1 - (2^{i-1}-1) \, </math>
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| Now, by setting <math> a = \frac{p_1 - 1}{2} </math>, we have <math> p_i = 2^{i} a + 1 </math>
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| Cunningham chains are also sometimes generalized to sequences of prime numbers (''p''<sub>1</sub>, ..., ''p''<sub>''n''</sub>) such that for all 1 ≤ ''i'' ≤ ''n'', ''p''<sub>''i''+1</sub> = ''ap''<sub>''i''</sub> + ''b'' for fixed [[coprime]] [[integer]]s ''a'', ''b''; the resulting chains are called '''generalized Cunningham chains'''.
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| A Cunningham chain is called '''complete''' if it cannot be further extended, i.e., if the previous or next term in the chain would not be a prime number anymore.
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| Examples of complete Cunningham chains of the first kind include these:
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| : 2, 5, 11, 23, 47 (The next number would be 95, but that is not prime.)
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| : 3, 7 (The next number would be 15, but that is not prime.)
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| : 29, 59 (The next number would be 119, but that is not prime.)
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| : 41, 83, 167 (The next number would be 335, but that is not prime.)
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| Examples of complete Cunningham chains of the second kind include these:
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| : 2, 3, 5 (The next number would be 9, but that is not prime.)
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| : 7, 13 (The next number would be 25, but that is not prime.)
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| : 19, 37, 73 (The next number would be 145, but that is not prime.)
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| : 31, 61 (The next number would be 121, but that is not prime.)
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| Cunningham chains are now considered useful in cryptographic systems since "they provide two concurrent suitable settings for the ElGamal cryptosystem ... [which] can be implemented in any field where the discrete logarithm problem is difficult."<ref>Joe Buhler, ''Algorithmic Number Theory: Third International Symposium, ANTS-III''. New York: Springer (1998): 290</ref>
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| == Largest known Cunningham chains ==
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| It follows from [[Dickson's conjecture]] and the broader [[Schinzel's hypothesis H]], both widely believed to be true, that for every ''k'' there are infinitely many Cunningham chains of length ''k''. There are, however, no known direct methods of generating such chains. | |
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| {| class="wikitable"
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| |+ Largest known Cunningham chain of length ''k'' (as of 14 December 2013<ref name="records">Dirk Augustin, [http://users.cybercity.dk/~dsl522332/math/Cunningham_Chain_records.htm ''Cunningham Chain records'']. Retrieved on 2013-12-14.</ref>)
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| |-
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| ! ''k'' !! Kind !! ''p''<sub>1</sub> (starting prime) !! Digits !! Year !! Discoverer
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| | 1 || || 2<sup>57885161</sup> − 1 || align="right" | 17425170 || 2013 || [[Curtis Cooper (mathematician)|Curtis Cooper]], [[Great Internet Mersenne Prime Search|GIMPS]]
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| |-
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| | rowspan="2" | 2 || 1st || 18543637900515×2<sup>666667</sup> − 1 || align="right" | 200701 || 2012 || Philipp Bliedung, [[PrimeGrid]]
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| | 2nd || 648309×2<sup>148310</sup> + 1 || align="right" | 44652 || 2010 || Tom Wu
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| | rowspan="2" | 3 || 1st || 914546877×2<sup>34772</sup> − 1 || align="right" | 10477 || 2010 || Tom Wu
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| |-
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| | 2nd || 82659189×2<sup>26997</sup> + 1 || align="right" | 8135 || 2010 || Tom Wu
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| | rowspan="2" | 4 || 1st || 1249097877×6599# − 1 || align="right" | 2835 || 2011 || Michael Angel
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| |-
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| | 2nd || 630698711×4933# + 1 || align="right" | 2105 || 2010 || Michael Angel
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| | rowspan="2" | 5 || 1st || 4250172704×2749# − 1 || align="right" | 1183 || 2012 || Dirk Augustin
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| | 2nd || 80670856865×2677# + 1 || align="right" | 1140 || 2011 || Michael Angel
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| | rowspan="2" | 6 || 1st || 37488065464×1483# − 1 || align="right" | 633 || 2010 || Dirk Augustin
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| |-
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| | 2nd || 37783362904×1097# + 1 || align="right" | 475 || 2006 || Dirk Augustin
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| | rowspan="2" | 7 || 1st || 162597166369×827# − 1 || align="right" | 356 || 2010 || Dirk Augustin
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| | 2nd || 668302064×593# + 786153598231 || align="right" | 251 || 2008 || Thomas Wolter & Jens Kruse Andersen
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| | rowspan="2" | 8 || 1st || 2×65728407627×431# − 1 || align="right" | 186 || 2005 || Dirk Augustin
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| | 2nd || 1148424905221×509# + 1 || align="right" | 224 || 2010 || Dirk Augustin
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| | rowspan="2" | 9 || 1st || 65728407627×431# − 1 || align="right" | 185 <!-- Do not replace this with anything less than 185 digits. Primecoin has a CC9 2nd kind record but this CC9 1st kind record is the overall CC9 record as of August 2013 --> || 2005 || Dirk Augustin
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| | 2nd || 182887101390961871050645934589918687746535370612015546956692154622371784133412186×223# + 1 || align="right" | 167 || 2013 || [[Primecoin]] ([http://primecoin.21stcenturymoneytalk.org/index.php?block_height=79349 block 79349])
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| | rowspan="2" | 10 || 1st || 44598464649019035883154084128331646888059795218766083584048621139159337786287845212160000×149# − 1 || align="right" | 146 || 2013 || Primecoin ([http://primecoin.21stcenturymoneytalk.org/index.php?block_height=182690 block 182690])
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| | 2nd || 2361221366027763072635481564211745987513418780208430432809344944242307595601310582×139# + 1 || align="right" | 137 || 2013 || Primecoin ([http://primecoin.21stcenturymoneytalk.org/index.php?block_height=125775 block 125775])
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| | rowspan="2" | 11 || 1st || 73853903764168979088206401473739410396455001112581722569026969860983656346568919×151# − 1 || align="right" | 140 || 2013 || Primecoin ([http://primecoin.21stcenturymoneytalk.org/index.php?block_height=95569 block 95569])
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| | 2nd || 8026337833619599372491948674562462668692014872229571339857384053514279156849912832×109# + 1 || align="right" | 127 || 2014 || Primecoin ([http://primecoin.21stcenturymoneytalk.org/index.php?block_height=365304 block 365304])
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| | rowspan="2" | 12 || 1st || 61592551716229060392971860549140211602858978086524024531871935735163762961673908480×71# − 1 || align="right" | 110 || 2013 || Primecoin ([http://primecoin.21stcenturymoneytalk.org/index.php?block_height=239833 block 239833])
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| | 2nd || 160433998429454286861864982184342218645773889300991352796925862298096263175269000×73# + 1 || align="right" | 109 || 2013 || Primecoin ([http://primecoin.21stcenturymoneytalk.org/index.php?block_height=323183 block 323183])
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| | rowspan="2" | 13 || 1st || 106680560818292299253267832484567360951928953599522278361651385665522443588804123392×61# − 1 || align="right" | 107 || 2014 || Primecoin ([http://primecoin.21stcenturymoneytalk.org/index.php?block_height=368051 block 368051])
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| | 2nd || 457905006824220038355933583213167637693837605663532362031057034656375497233892960×37# + 1 || align="right" | 94 || 2014 || Primecoin ([http://primecoin.21stcenturymoneytalk.org/index.php?block_height=375981 block 375981])
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| | rowspan="2" | 14 || 1st || 9510321949318457733566099 || align="right" | 25 || 2004 || Jens Kruse Andersen
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| | 2nd || 335898524600734221050749906451371 || align="right" | 33 || 2008 || Jens Kruse Andersen
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| | rowspan="2" | 15 || 1st || 662311489517467124375039 || align="right" | 24 || 2008 || Jaroslaw Wroblewski
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| | 2nd || 28320350134887132315879689643841 || align="right" | 32 || 2008 || Jaroslaw Wroblewski
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| | rowspan="2" | 16 || 1st || 91304653283578934559359 || align="right" | 23 || 2008 || Jaroslaw Wroblewski
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| | 2nd || 2368823992523350998418445521 || align="right" | 28 || 2008 || Jaroslaw Wroblewski
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| | rowspan="2" | 17 || 1st || 2759832934171386593519 || align="right" | 22 || 2008 || Jaroslaw Wroblewski
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| |-
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| | 2nd || 1302312696655394336638441 || align="right" | 25 || 2008 || Jaroslaw Wroblewski
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| |}
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| ''q''# denotes the [[primorial]] 2×3×5×7×...×''q''.
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| {{As of|2013}}, the longest known Cunningham chain of either kind is of length 17. The first known was of the 1st kind starting at 2759832934171386593519, discovered by Jaroslaw Wroblewski in 2008 where he also found some of the 2nd kind.<ref name="records"/>
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| == Congruences of Cunningham chains ==
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| Let the odd prime <math>p_1</math> be the first prime of a Cunningham chain of the first kind. The first prime is odd, thus <math>p_1 \equiv 1 \pmod{2}</math>. Since each successive prime in the chain is <math>p_{i+1} = 2p_i + 1</math> it follows that <math>p_i \equiv 2^i - 1 \pmod{2^i}</math>. Thus, <math>p_2 \equiv 3 \pmod{4}</math>, <math>p_3 \equiv 7 \pmod{8}</math>, and so forth.
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| The above property can be informally observed by considering the primes of a chain in base 2. (Note that, as with all bases, multiplying by the number of the base "shifts" the digits to the left.) When we consider <math>p_{i+1} = 2p_i + 1</math> in base 2, we see that, by multiplying <math>p_i</math> by 2, the least significant digit of <math>p_i</math> becomes the secondmost least significant digit of <math>p_{i+1}</math>. Because <math>p_i</math> is odd—that is, the least significant digit is 1 in base 2--we know that the secondmost least significant digit of <math>p_{i+1}</math> is also 1. And, finally, we can see that <math>p_{i+1}</math> will be odd due to the addition of 1 to <math>2p_i</math>. In this way, successive primes in a Cunningham chain are essentially shifted left in binary with ones filling in the least significant digits. For example, here is a complete length 6 chain which starts at 141361469:
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| {| border="1" align="center" class="wikitable"
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| ! Binary || Decimal
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| |- align="right"
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| | 1000011011010000000100111101 || 141361469
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| |- align="right"
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| | 10000110110100000001001111011 || 282722939
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| |- align="right"
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| | 100001101101000000010011110111 || 565445879
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| |- align="right"
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| | 1000011011010000000100111101111 || 1130891759
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| |- align="right"
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| | 10000110110100000001001111011111 || 2261783519
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| |- align="right"
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| | 100001101101000000010011110111111 || 4523567039
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| |}
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| A similar result holds for Cunningham chains of the second kind. From the observation that <math>p_1 \equiv 1 \pmod{2}</math> and the relation <math>p_{i+1} = 2 p_i - 1</math> it follows that <math>p_i \equiv 1 \pmod{2^i}</math>. In binary notation, the primes in a Cunningham chain of the second kind end with a pattern "0...01", where, for each <math>i</math>, the number of zeros in the pattern for <math>p_{i+1}</math> is one more than the number of zeros for <math>p_i</math>. As with Cunningham chains of the first kind, the bits left of the pattern shift left by one position with each successive prime.
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| Similarly, because <math> p_i = 2^{i-1}p_1 + (2^{i-1}-1) \, </math> it follows that <math>p_i \equiv 2^{i-1} - 1 \pmod{p_1}</math>. But, by [[Fermat's little theorem]], <math>2^{p_1-1} \equiv 1 \pmod{p_1}</math>, so <math>p_1</math> divides <math>p_{p_1}</math> (i.e. with <math> i = p_1 </math>). Thus, no Cunningham chain can be of infinite length.<ref>{{cite journal|last=Löh|first=Günter|title=Long chains of nearly doubled primes|journal=Mathematics of Computation|year=1989|month=October|volume=53|issue=188|pages=751–759|doi=10.1090/S0025-5718-1989-0979939-8|url=http://www.ams.org/journals/mcom/1989-53-188/S0025-5718-1989-0979939-8/}}</ref>
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| ==References==
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| <references/>
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| == See also ==
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| * [[Primecoin]], which uses Cunningham chains as a proof-of-work system
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| == External links ==
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| * [http://primes.utm.edu/glossary/page.php?sort=CunninghamChain The Prime Glossary: Cunningham chain]
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| * [http://primes.utm.edu/links/theory/special_forms/Cunningham_chains/ PrimeLinks++: Cunningham chain]
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| * [http://oeis.org/A005602 Sequence A005602] in [[OEIS]]: the first term of the lowest complete Cunningham chains of the first kind of length ''n'', for 1 ≤ ''n'' ≤ 14
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| * [http://oeis.org/A005603 Sequence A005603] in [[OEIS]]: the first term of the lowest complete Cunningham chains of the second kind with length ''n'', for 1 ≤ ''n'' ≤ 15
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| {{Prime number classes}}
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| [[Category:Prime numbers]]
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