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Cunningham chain
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== Largest known Cunningham chains == 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. There are computing competitions for the longest Cunningham chain or for the one built up of the largest primes, but unlike the breakthrough of [[Ben J. Green]] and [[Terence Tao]] β the [[GreenβTao theorem]], that there are arithmetic progressions of primes of arbitrary length β there is no general result known on large Cunningham chains to date. {| class="wikitable sortable" |+ Largest known Cunningham chain of length ''k'' (as of 18 February 2025<ref name="records">Norman Luhn & Dirk Augustin, [https://www.pzktupel.de/JensKruseAndersen/cc.htm''Cunningham Chain records'']. Retrieved on 2025-02-18.</ref>) |- ! ''k'' !! Kind !! ''p''<sub>1</sub> (starting prime) !! Digits !! Year !! Discoverer |- | 1 || 1st / 2nd || 2<sup>136279841</sup> β 1 || align="right" | 41024320 || 2024 || Luke Durant, [[Great Internet Mersenne Prime Search|GIMPS]] |- | rowspan="2" | 2 || 1st || 2618163402417Γ2<sup>1290000</sup> β 1 || align="right" | 388342 || 2016 || [[PrimeGrid]] |- | 2nd || 213778324725Γ2<sup>561417</sup> + 1 || align="right" | 169015 || 2023 || Ryan Propper & Serge Batalov |- | rowspan="2" | 3 || 1st || 1128330746865Γ2<sup>66439</sup> β 1 || align="right" | 20013 || 2020 || Michael Paridon |- | 2nd || 214923707595Γ2<sup>49073</sup> + 1 || align="right" | 14784 || 2025 || Serge Batalov |- | rowspan="2" | 4 || 1st || 93003628384Γ10111# β 1 || align="right" | 4362 || 2025 || Serge Batalov |- | 2nd || 49325406476Γ9811# + 1 || align="right" | 4234|| 2019 || Oscar Γstlin |- | rowspan="2" | 5 || 1st || 475676794046977267Γ4679# β 1 || align="right" | 2019 || 2024 || Andrey Balyakin |- | 2nd || 181439827616655015936Γ4673# + 1 || align="right" | 2018 || 2016 || Andrey Balyakin |- | rowspan="2" | 6 || 1st || 2799873605326Γ2371# β 1 || align="right" | 1016 || 2015 || Serge Batalov |- | 2nd || 37015322207094Γ2339# + 1 || align="right" | 1001 || 2025 || Serge Batalov |- | rowspan="2" | 7 || 1st || 82466536397303904Γ1171# β 1 || align="right" | 509 || 2016 || Andrey Balyakin |- | 2nd || 25802590081726373888Γ1033# + 1 || align="right" | 453 || 2015 || Andrey Balyakin |- | rowspan="2" | 8 || 1st || 89628063633698570895360Γ593# β 1 || align="right" | 265 || 2015 || Andrey Balyakin |- | 2nd || 2373007846680317952Γ761# + 1 || align="right" | 337 || 2016 || Andrey Balyakin |- | rowspan="2" | 9 || 1st || 553374939996823808Γ593# β 1 || align="right" | 260 || 2016 || Andrey Balyakin |- | 2nd || 173129832252242394185728Γ401# + 1 || align="right" | 187 || 2015 || Andrey Balyakin |- | rowspan="2" | 10 || 1st || 3696772637099483023015936Γ311# β 1 || align="right" | 150 || 2016 || Andrey Balyakin |- | 2nd || 2044300700000658875613184Γ311# + 1 || align="right" | 150 || 2016 || Andrey Balyakin |- | rowspan="2" | 11 || 1st || 73853903764168979088206401473739410396455001112581722569026969860983656346568919Γ151# β 1 || align="right" | 140 || 2013 || Primecoin ([https://primes.zone/#records block 95569]) |- | 2nd || 341841671431409652891648Γ311# + 1 || align="right" | 149 || 2016 || Andrey Balyakin |- | rowspan="2" | 12 || 1st || 288320466650346626888267818984974462085357412586437032687304004479168536445314040Γ83# β 1 || align="right" | 113 || 2014 || Primecoin ([https://primes.zone/#records block 558800]) |- | 2nd || 906644189971753846618980352Γ233# + 1 || align="right" | 121 || 2015 || Andrey Balyakin |- | rowspan="2" | 13 || 1st || 106680560818292299253267832484567360951928953599522278361651385665522443588804123392Γ61# β 1 || align="right" | 107 || 2014 || Primecoin ([https://primes.zone/#records block 368051]) |- | 2nd || 38249410745534076442242419351233801191635692835712219264661912943040353398995076864Γ47# + 1 || align="right" | 101 || 2014 || Primecoin ([https://primes.zone/#records block 539977]) |- | rowspan="2" | 14 || 1st || 4631673892190914134588763508558377441004250662630975370524984655678678526944768Γ47# β 1 || align="right" | 97 || 2018 || Primecoin ([https://primes.zone/#records block 2659167]) |- | 2nd || 5819411283298069803200936040662511327268486153212216998535044251830806354124236416Γ47# + 1 || align="right" | 100 || 2014 || Primecoin ([https://primes.zone/#records block 547276]) |- | rowspan="2" | 15 || 1st || 14354792166345299956567113728Γ43# - 1 || align="right" | 45 || 2016 || Andrey Balyakin |- | 2nd || 67040002730422542592Γ53# + 1 || align="right" | 40 || 2016 || Andrey Balyakin |- | rowspan="2" | 16 || 1st || 91304653283578934559359 || align="right" | 23 || 2008 || Jaroslaw Wroblewski |- | 2nd || 2Γ1540797425367761006138858881 β 1 || align="right" | 28 || 2014 || Chermoni & Wroblewski |- | rowspan="2" | 17 || 1st || 2759832934171386593519 || align="right" | 22 || 2008 || Jaroslaw Wroblewski |- | 2nd || 1540797425367761006138858881 || align="right" | 28 || 2014 || Chermoni & Wroblewski |- | 18 || 2nd || 658189097608811942204322721 || align="right" | 27 || 2014 || Chermoni & Wroblewski |- | 19 || 2nd || 79910197721667870187016101 || align="right" | 26 || 2014 || Chermoni & Wroblewski |} ''q''# denotes the [[primorial]] 2 Γ 3 Γ 5 Γ 7 Γ ... Γ ''q''. {{As of|2018}}, the longest known Cunningham chain of either kind is of length 19, discovered by Jaroslaw Wroblewski in 2014.<ref name="records"/>
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