At this site I have collected together all the largest known examples of certain types of dense clusters of prime numbers. The idea is to generalise the notion of prime twins - pairs of prime numbers {p, p + 2} - to groups of three or more.
Prepared by Tony Forbes; anthony.d.forbes@gmail.com.
Site address: http://anthony.d.forbes.googlepages.com/ktuplets.htm.
1 Jun 2021
Prime 20-tuplet (NEW RECORD!)
1236637204227022808686214288579 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (31 digits, May 23, 2021, Raanan Chermoni & Jaroslaw Wroblewski)
Prime 9-tuplets (INCLUDING NEW RECORD!)
171261 40574427698443038292751799032279392120235175186111 33550038875063021572377776139492086991109134537768 * 421# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (277 digits, 29 May 2021, XpoolX)
66160221840972226140223807708905566890407723048320 78213536970036538738519759031878588550223823046887 * 409# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (267 digits, 22 May 2021, XpoolX)
60 06559904350134849252155379950254909378835186594558 86375786941368048031982994793132628369636851519027 * 401# + 701889794782061 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (266 digits, 01 May 2021, Suprnova)
218 35722977202308313590097241777421011879112635152949 76358043264898170326365943244347564793366086140266 * 397# + 1277156391416021 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (264 digits, 17 May 2021, PrimaPool)
3483 87853121458649008179383696133766318175683929635059 02067913510453141649847518582456424344347357754742 * 383# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (260 digits, 23 Apr 2021, ric1qmstmv36kuthe7xu4dxzhz6jwnk27t4xqtvmqfa)
244819757603183304556386786638379528596212689 91918113342032150939302877507639213478752614572807 * 401# + 226193845148629 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 (258 digits, 09 May 2021, Pttn)
6309 21912327808796744952939633997746226000817332256293 31799182311085295909231723068059035832331092803319 * 379# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (258 digits, 06 May 2021, XpoolX)
3050506767208880140795226836760067306733856274919 79965295066581348929079873313258875248946926733351 * 383# + 380284918609481 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (255 digits, 06 May 2021, PrimaPool)
69638 39794595034657927780534260078976037788331435278665 70574855127452541573990000208963455191492119007079 * 359# + 701889794782061 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (251 digits, 16 Apr 2021, TomB)
Prime 8-tuplets
5602666 54113573653807256039421159095716686744230887487823 55514929545904535010555909219571677274687952903198 * 431# + 1418575498583 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 (281 digits, 29 May 2021, PrimaPool)
49213 11217401936811103526156352113727870867374033023319 46846916505063641420547506943834266263728517203001 * 431# + 235520915372201 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (279 digits, 29 May 2021, ric1qdekpf4v3lwhfu2apasq4metfuuu35q4keu5prq)
1727317194 68357534695186802566959090837823806487250092096759 10868709024745855633473508971497256694964580562010 * 419# + 226374233346623 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 (279 digits, 29 May 2021, PrimaPool)
7972898380320551572902687408841330128076444 69800381838535408958174313617814919077117774769636 * 449# + 114189340938131 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (278 digits, 25 May 2021, Pttn)
190691452886 60724075207124034559127904455846734774801103911151 31600178350904532358406293408041601503459947335601 * 409# + 226554621544613 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 (278 digits, 31 May 2021, PrimaPool)
336156 34685209502103299496432743672107320096495787519596 55486658770545087397881480698253776672526729812074 * 421# + 114023297140211 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (277 digits, 25 May 2021, PrimaPool)
43414128754 21363078855220435892914409488271998218360408303523 02979909931330843256806773392894057481877865551107 * 409# + 226554621544613 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 (277 digits, 29 May 2021, XpoolX)
Many more
The smallest 3000, 4000 and 5000 digit proven prime triplets. Certificates was uploaded to factordb.com
10^2999 + 25740029131 + d, d = 0, 2, 6 (3000 digits, May 2021, Norman Luhn)
10^2999 + 37274603937 + d, d = 0, 4, 6 (3000 digits, May 2021, Norman Luhn)
10^3999 + 182402621497 + d, d = 0, 2, 6 (4000 digits, May 2021, Norman Luhn)
10^3999 + 243095638113 + d, d = 0, 4, 6 (4000 digits, May 2021, Norman Luhn)
10^4999 + 70852892827 + d, d = 0, 2, 6 (5000 digits, May 2021, Norman Luhn)
10^4999 + 244793127627 + d, d = 0, 4, 6 (5000 digits, May 2021, Norman Luhn)
14 Apr 2021
Prime 9-tuplets (INCLUDING NEW RECORD!)
2215574667032155520 01687324835590414397820789810764345732668494831717 17294207993515336980845763232977257432862782607440 * 313# + 855709 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 (247 digits, 10 Apr 2021, ric1quau6a3z8qu4ar204pwgz2vdndyta455vsn99lq)
15768789 82591682476593228646303308772443630065692903753627 93747224209093455299633188624996764587832200036071 * 347# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (246 digits, 10 Apr 2021, ric1qz4cjxt2vvje35mfk85tdl3lf3xffq7p70admfm)
2511580414961521073463077876332682567182843927 20151793261635788053028333576682615819341450443970 * 347# + 855709 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 (234 digits, 30 Mar 2021, ric1qp8w3x4nzq6fg5s3nk8r6hkrwcq3sr38tw6scs3)
Prime 8-tuplets
566957934 38775675886628672818286249081064597579487749354717 29173742601730954612966130436450660208024251652599 * 367# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (257 digits, 05 Apr 2021, ric1quau6a3z8qu4ar204pwgz2vdndyta455vsn99lq)
30831605 03765710417063455157626076315815440183653905359568 50877877792067318698516316134777684103680311943675 * 367# + 1146773 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 (256 digits, 05 Apr 2021, ric1quau6a3z8qu4ar204pwgz2vdndyta455vsn99lq)
38136 09649621240197693419538656539466535439309514373505 84419561374060339273913731376431546301651648046520 62678870255864628344457752757987524975076311301549 03939752506454993046720152330164438305466899311878 43358297347212784087050308628262234355875989849971 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (255 digits, 05 Apr 2021, ric1quau6a3z8qu4ar204pwgz2vdndyta455vsn99lq)
5034 72008854601692807730675755250313167673397053914977 79308581428034944822699756866737727164657677695480 89213162971784440228468271476069842988436925888061 01899804078840413921929420209546556452133586910641 18874830805258955379364823114679157522511533435621 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (254 digits, 04 Apr 2021, ric1quau6a3z8qu4ar204pwgz2vdndyta455vsn99lq)
31983853 69199138078246569309313483800470807757855953330189 45738911665531345947322559229245533058585378416388 * 359# + 74266253 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 (254 digits, 07 Apr 2021, ric1q47qw76nrc606ueak5m4wzms9u87ueftc6lsesp)
10377634 58082011454697895277152308833206663100581339162091 08197252370462757239210706422204771544111190002409 * 359# + 1277156391416021 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (253 digits, 05 Apr 2021, ric1qp8w3x4nzq6fg5s3nk8r6hkrwcq3sr38tw6scs3)
3227383071 98823814140091443207368864240414544015487058829844 39658156560846662543890209688994888617070014513312 * 353# + 3360877662097841 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (253 digits, 04 Apr 2021, ric1qf9defee7d4m4vadfrfjfy8htdtr5ds9023hwyq)
12704 72419625254415397900339456930862505586455471086519 82008615478003292329382327312421624888945846506192 * 367# + 380284918609481 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (253 digits, 13 Apr 2021, RSoTqUcauJewHZYmPnM8w3p1LVcV5C2Fh1)
29 55477519628442054728539781356568740251579139483773 53705287862391139125839802773942920435931881509866 * 373# + 1146773 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 (253 digits, 06 Apr 2021, ric1quau6a3z8qu4ar204pwgz2vdndyta455vsn99lq)
10246 45975923245512436849213582119622351320754463763171 74301840042129912947563118664521065080168942747445 * 367# + 2325810733931801 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (253 digits, 06 Apr 2021, ric1qkzm5qfppw8n7m404j9t3t22xkvsjf39rfsjl7x)
Many more
14 Mar 2021
Prime 8-tuplet (NEW WORLD RECORD!)
6879356578124627875380298699944709053335 * 677# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (324 digits, 12 Mar 2021, Michalis Christou)
Prime 12-tuplets (Smallest with 50 digits)
10000000000000000000000000000929532973818094710897 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (50 digits, 24 Feb 2021, Norman Luhn)
10000000000000000000000000000896396147387349765031 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (50 digits, 24 Feb 2021, Norman Luhn)
Prime 13-tuplets (Smallest with 40 digits)
1000000000000000002713562652524314606953 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48 (40 digits, 10 Mar 2021, Norman Luhn)
1000000000000000002334523699629280598673 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36, 46, 48 (40 digits, 10 Mar 2021, Norman Luhn)
1000000000000000000368816080526066037739 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (40 digits, 10 Mar 2021, Norman Luhn)
1000000000000000000349508508460276218891 + d, d = 0, 6, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 48 (40 digits, 10 Mar 2021, Norman Luhn)
1000000000000000000349508508460276218889 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (40 digits, 10 Mar 2021, Norman Luhn)
1000000000000000000282197071067938130221 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (40 digits, 10 Mar 2021, Norman Luhn)
Prime 14-tuplet (obtained by combining the 4th and 5th Prime 13-tuplets, above)
1000000000000000000349508508460276218889 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (40 digits, 10 Mar 2021, Norman Luhn)
05 Feb 2021
Prime 7-tuplet (NEW RECORD, FIRST WITH MORE THAN 1000 DIGITS!)
113225039190926127209 * 2339# / 57057 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20 (1002 digits, 27 Jan 2021, Peter Kaiser)
Prime 17-tuplet (NEW RECORD!)
150048143328514263089612453401301 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 (33 digits, Feb 2021, Roger Thompson)
Prime 16-tuplet
302458608131364933637125192102583 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (33 digits, Feb 2021, Roger Thompson)
Prime 13-tuplets (Smallest with 35 digits)
10000000000000325778825790175217703 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36, 46, 48 (35 digits, Feb 2021, Norman Luhn)
10000000000000324000701496110723931 + d, d = 0, 6, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 48 (35 digits, Feb 2021, Norman Luhn)
10000000000000108412629077454977119 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (35 digits, Feb 2021, Norman Luhn)
10000000000000094989640220894283993 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48 (35 digits, Feb 2021, Norman Luhn)
10000000000000054122451329461300669 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (35 digits, Feb 2021, Norman Luhn)
10000000000000015141548551355951851 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (35 digits, Feb 2021, Norman Luhn)
Prime 14-tuplets (Smallest with 35 digits)
10000000000001275924044876917671361 + d, d = 00, 02, 06, 08, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (35 digits, Feb 2021, Norman Luhn)
10000000000009283441665311798539399 + d, d = 00, 02, 08, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (35 digits, Feb 2021, Norman Luhn)
Prime 16-tuplets (Smallest with 25 digits)
1015074281315414986743013 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (25 digits, Feb 2021, Norman Luhn)
1008037335701436528651167 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 (25 digits, Feb 2021, Norman Luhn)
23 Jan 2021
Prime 5-tuplet (NEW RECORD!)
566761969187 * 4733#/2 + d, d = −8, −4, −2, 2, 4 (2034 digits, December 2020, Serge Batalov, NEWPGEN, OPENPFGW, PRIMO)
Prime 20-tuplet (NEW RECORD!)
1188350591359110800209379560799 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (31 digits, January 21, 2021, Raanan Chermoni & Jaroslaw Wroblewski)
Prime 8-tuplets (Smallest with 150 digits)
10^149 + 177107310312127411 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (150 digits, January 2021, Norman Luhn)
10^149 + 883945334707753267 + d, d = 0, 2, 6, 12, 14, 20, 24, 26 (150 digits, January 2021, Norman Luhn)
10^149 + 935628779313782743 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 (150 digits, January 2021, Norman Luhn)
31 Dec 2020
Prime 20-tuplet (NEW RECORD!)
1153897621507935436463788957529 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (31 digits, December 26, 2020, Raanan Chermoni & Jaroslaw Wroblewski)
Prime 8-tuplet (Smallest with 200 digits and pattern {0, 6, 8, 14, 18, 20, 24, 26})
10^199 + 4456720213751803153 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 (200 digits, December 2020, Norman Luhn)
Prime 8-tuplet (Smallest with 200 digits and pattern {0, 2, 6, 12, 14, 20, 24, 26})
10^199 + 589262946758538727 + d, d = 0, 2, 6, 12, 14, 20, 24, 26 (200 digits, December 2020, Norman Luhn)
22 Dec 2020
Prime 20-tuplet (NEW RECORD!)
1135540756371356698957890225821 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (31 digits, December 19, 2020, Raanan Chermoni & Jaroslaw Wroblewski)
Prime 8-tuplet (Smallest with 200 digits and pattern {0, 2, 6, 8, 12, 18, 20, 26})
10^199 + 4342765936145019181 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (200 digits, December 2020, Norman Luhn)
Prime numbers are the building blocks of arithmetic. They are a special type of number because they cannot be broken down into smaller factors. 13 is prime because 13 is 1 times 13 (or 13 times 1), and that's it. There's no other way of expressing 13 as something times something. On the other hand, 12 is not prime because it splits into 2 times 6, or 3 times 4.
The first prime is 2. The next is 3. Then 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503 and so on. If you look at the first 10000 primes, you will see a list of numbers with no obvious pattern. There is even an air of mystery about them; if you didn't know they were prime numbers, you would probably have no idea how to continue the sequence. Indeed, if you do manage discover a simple pattern, you will have succeeded where some of the finest brains of all time have failed. For this is an area where mathematicians are well and truly baffled.
We do know fair amount about prime numbers, and an excellent starting point if you want to learn more about the subject is Chris Caldwell's web site: The Largest Known Primes. We know that the sequence of primes goes on for ever. We know that it thins out. The further you go, the rarer they get. We even have a simple formula for estimating roughly how many primes there are up to some large number without having to count them one by one. However, even though prime numbers have been the object of intense study by mathematicians for hundreds of years, there are still fairly basic questions which remain unanswered.
If you look down the list of primes, you will quite often see two consecutive odd numbers, like 3 and 5, 5 and 7, 11 and 13, 17 and 19, or 29 and 31. We call these pairs of prime numbers {p, p + 2} prime twins.
The evidence suggests that, however far along the list of primes you care to look, you will always eventually find more examples of twins. Nevertheless, - and this may come as a surprise to you - it is not known whether this is in fact true. Possibly they come to an end. But it seems more likely that - like the primes - the sequence of prime twins goes on forever. However, Mathematics has yet to provide a rigorous proof.
One of the things mathematicians do when they don't understand something is produce bigger and better examples of the objects that are puzzling them. We run out of ideas, so we gather more data - and this is just what we are doing at this site; if you look ahead to section 2, you will see that I have collected together the ten largest known prime twins.
If you search the list for triples of primes {p, p + 2, p + 4}, you will not find very many. In fact there is only one, {3, 5, 7}, right at the beginning. And it's easy to see why. As G. H. Hardy & J. E. Littlewood observed [HL22], at least one of the three is divisible by 3.
Obviously it is asking too much to squeeze three primes into a range of four. However, if we increase the range to six and look for combinations {p, p + 2, p + 6} or {p, p + 4, p + 6}, we find plenty of examples, beginning with {5, 7, 11}, {7, 11, 13}, {11, 13, 17}, {13, 17, 19}, {17, 19, 23}, {37, 41, 43}, .... These are what we call prime triplets, and one of the main objectives of this site is to collect together all the largest known examples. Just as with twins, it is believed - but not known for sure - that the sequence of prime triplets goes on for ever.
Similar considerations apply to groups of four, where this time we require each of {p, p + 2, p + 6, p + 8} to be prime. Once again, it looks as if they go on indefinitely. The smallest is {5, 7, 11, 13}. We don't count {2, 3, 5, 7} even though it is a denser grouping. It is an isolated example which doesn't fit into the scheme of things. Nor, for more technical reasons, do we count {3, 5, 7, 11}.
The sequence continues with {11, 13, 17, 19}, {101, 103, 107, 109}, {191, 193, 197, 199}, {821, 823, 827, 829}, .... The usual name is prime quadruplets, although I have also seen the terms full house, inter-decal prime quartet (!) and prime decade - a reference to the pattern made by their decimal digits. All primes greater than 5 end in one of 1, 3, 7 or 9, and the four primes in a (large) quadruplet always occur in the same ten-block. Hence there must be exactly one with each of these unit digits. And just to illustrate the point, here is another example; the smallest prime quadruplet of 50 digits, found by G. John Stevens in 1995 [S95]:
10000000000000000000000000000000000000000058537891,
10000000000000000000000000000000000000000058537893,
10000000000000000000000000000000000000000058537897,
10000000000000000000000000000000000000000058537899.
We can go on to define prime quintuplets, sextuplets, septuplets, octuplets, nonuplets, and so on. I had to go to the full Oxford English Dictionary for the last one - the Concise Oxford jumps from 'octuplets' to 'decuplets'. The OED also defines 'dodecuplets', but apparently there are no words for any of the others. Presumably I could make them up, but instead I shall use the number itself when I want to refer to, for example, prime 11-tuplets. I couldn't find the general term 'k-tuplets' in the OED either, but it is the word that seems to be in common use by the mathematical community.
For now, I will define a prime k-tuplet as a sequence of consecutive prime numbers such that the distance between the first and the last is in some sense as small as possible. If you think I am being too vague, there is a more precise definition later on.
At this site I have collected together what I believe to be the largest known prime k-tuplets for k = 2, 3, 4, ..., 20 and 21. I do not know of any prime k-tuplets for k greater than 21, except for the ones that occur near the beginning of the prime number sequence.
Multiplication is often denoted by an asterisk: x*y is x times y.
For k > 2, the somewhat bizarre notation N + b1, b2, ..., bk is used (only in linked pages) to denote the k numbers {N + b1, N + b2, ..., N + bk}.
Prime twins are represented as N ± 1, which is short for N plus one and N minus one.
I also use the notation n# of Caldwell and Dubner [CD93] as a convenient shorthand for the product of all the primes less than or equal to n. Thus, for example, 20# = 2*3*5*7*11*13*17*19 = 9699690.
I would like to keep this site as up to date as possible. Therefore, can I urge you to please send any new, large prime k-tuplets to me. You can see what I mean by 'large' by studying the lists. If the numbers are not too big, say up to 500 digits, I am willing to double-check them myself. Otherwise I would appreciate some indication of how you proved that your numbers are true primes. Email address: anthony.d.forbes@gmail.com.
2996863034895 * 21290000 ± 1 (388342 digits, Sep 2016, Tom Greer, TWINGEN, PRIMEGRID, LLR)
3756801695685 * 2666669 ± 1 (200700 digits, Dec 2011, Timothy Winslow, TWINGEN, PRIMEGRID, LLR)
65516468355 * 2333333 ± 1 (100355 digits, Aug 2009, Peter Kaiser, NEWGEN, PRIMEGRID, TPS, LLR)
12770275971 * 2222225 ± 1 (66907 digits, Jul 2017, Bo Tornberg, TWINGEN, LLR TWIN)
70965694293 * 2200006 ± 1 (60219 digits, Apr 2016, S. Urushihata)
66444866235 * 2200003 ± 1 (60218 digits, Apr 2016, S. Urushihata)
4884940623 * 2198800 ± 1 (59855 digits, Jul 2015, Kwok, PSIEVE, LLR)
2003663613 * 2195000 ± 1 (58711 digits, Jan 2007, Eric Vautier, Dmitri Gribenko, Patrick W. McKibbon, Michaek Kwok, Andrea Pacini, Rytis Slatkevicius)
191547657 * 2173372 ± 1 (52199 digits, Nov 2020, Stephan Schöler Eric Vautier, MULTISIEVE, LLR)
38529154785 * 2173250 ± 1 (52165 digits, Jul 2014, Serge Batalov, NEWPGEN, LLR)
See Chris Caldwell, The Largest Known Primes for further (and possibly more up to date) information.
4111286921397 * 266420 + d, d = −1, 1, 5 (20008 digits, 24 Apr 2019, Peter Kaiser, POLYSIEVE, LLR, PRIMO)
6521953289619 * 255555 + d, d = −5, −1, 1 (16737 digits, Apr 2013, Peter Kaiser)
3221449497221499 * 234567 + d, d = −1, 1, 5 (10422 digits, Sep 2015, Peter Kaiser, NEWGEN, LLR, PRIMO5)
1288726869465789 * 234567 + d, d = −5, −1, +1 (10421 digits, Apr 2014, Peter Kaiser)
647935598824239 * 233619 + d, d = −1, 1, 5 (10136 digits, 22 May 2019, Peter Kaiser, PRIMO)
209102639346537 * 233620 + d, d = −1, 1, 5 (10135 digits, 22 May 2019, Peter Kaiser, PRIMO)
185353103135997 * 233620 + d, d = −1, 1, 5 (10135 digits, 22 May 2019, Peter Kaiser, PRIMO)
162615027598677 * 233620 + d, d = −1, 1, 5 (10135 digits, 22 May 2019, Peter Kaiser, PRIMO)
667674063382677 * 233608 + d, d = 1, 5, 7 (10132 digits, 27 Feb 2019, Peter Kaiser, PRIMO)
667674063382677 * 233608 + d, d = −1, 1, 5 (10132 digits, 27 Feb 2019, Peter Kaiser, PRIMO)
667674063382677 * 233608 + d, d = −1, 1, 5, 7 (10132 digits, 27 Feb 2019, Peter Kaiser, PRIMO)
4122429552750669 * 216567 + d, d = −1, 1, 5, 7 (5003 digits, Mar 2016, Peter Kaiser, GSIEVE, NewPGen, LLR, PRIMO)
2673092556681 * 153048 + d, d = −4, −2, 2, 4 (3598 digits, Sep 2015, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)
2339662057597 * 103490 + d, d = 1, 3, 7, 9 (3503 digits, Dec 2013, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)
305136484659 * 211399 + d, d = −1, 1, 5, 7 (3443 digits, Sep 2013, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)
722047383902589 * 211111 + d, d = −1, 1, 5, 7 (3360 digits, Apr 2013, Reto Keiser, NEWPGEN, PFGW, PRIMO)
43697976428649 * 29999 + d, d = −1, 1, 5, 7 (3024 digits, Mar 2012, Peter Kaiser)
46359065729523 * 28258 + d, d = −1, 1, 5, 7 (2500 digits, Nov 2011, Reto Keiser, NEWPGEN, PFGW, PRIMO)
1367848532291 * 5591# / 35 + d, d = −1, 1, 5, 7 (2401 digits, Aug 2011, Norman Luhn, NEWPGEN, PFGW, PRIMO)
25796119248 * 4987# / 35 + d, d = −1, 1, 5, 7 (2135 digits, May 2011, Gary Chaffey)
566761969187 * 4733#/2 + d, d = −8, −4, −2, 2, 4 (2034 digits, December 2020, Serge Batalov, NEWPGEN, OPENPFGW, PRIMO)
126831252923413 * 4657# / 273 + d, d = 1, 3, 7, 9, 13 (2002 digits, 8 Nov 2020, Peter Kaiser, PRIMO)
394254311495 * 3733# / 2 + d, d = -8, -4, -2, 2, 4 (1606 digits, Nov 2017, Serge Batalov, NEWPGEN, OPENPFGW, PRIMO)
2316765173284 * 3600# + 16061 + d, d = 0, 2, 6, 8, 12 (1543 digits, 16 Oct 2016, Norman Luhn, PRIMO)
163252711105 * 3371# / 2 + d, d = −8, −4, −2, 2, 4 (1443 digits, Jan 2014, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)
9039840848561 * 3299# / 35 + d, d = −5, −1, 1, 5, 7 (1401 digits, Dec 2013, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)
699549860111847 * 24244 + d, d = −1, 1, 5, 7, 11 (1293 digits, Dec 2013, Reto Keiser, R. Gerbicz, PFGW, PRIMO)
405095429109490796 * 2683# + 16057 + d, d = 0, 4, 6, 10, 12 (1150 digits, 4 Jul 2020, Michael Bell, RIEMINER, ECPP-DJ)
566650659276 * 2621# + 1615841 + d, d = 0, 2, 6, 8, 12 (1117 digits, Dec 2013, David Broadhurst, PRIMO, OpenPFGW)
554729409262 * 2621# + 1615841 + d, d = 0, 2, 6, 8, 12 (1117 digits, Dec 2013, David Broadhurst, PRIMO, OpenPFGW)
28993093368077 * 2400# + 19417 + d, d = 0, 4, 6, 10, 12, 16 (1037 digits, 14 Mar 2016, Norman Luhn, APSIEVE, PRIMO)
6646873760397777881866826327962099685830865900246688640856 * 1699# + 43777 + d, d = 0, 4, 6, 10, 12, 16 (780 digits, 8 Nov 2018, Vidar Nakling, PRIMO)
29720510172503062360713760607985203309940766118866743491802189150471978534404249 * 22299 + 14271253084334081637544486111223831073612730979632132919368177563415768349505 + d, d = 0, 4, 6, 10, 12, 16 (772 digits, 1/28/2018, Riecoin #822096)
29749903422302373222996698880833194129159047179535887991184960156219652236318921 * 22293 + 679631792885016654160023247517239700227428004849763556497260661860592843345 + d, d = 0, 4, 6, 10, 12, 16 (770 digits, 12/9/2017, Riecoin #793872)
29696802688480280387313212926526693549449146292085717645262228449092881114972806 * 22290 + 1946690158750077943506249776690378666457458353296002764327070450442847661633 + d, d = 0, 4, 6, 10, 12, 16 (769 digits, 2/25/2018, Riecoin #838224)
29744205023784420961031622414734790416939049568996819659808238403983863222665068 * 22288 + 14305894933680691041378655981062938998356035914288745998258984615535179477709 + d, d = 0, 4, 6, 10, 12, 16 (769 digits, 2/18/2018, Riecoin #834192)
29707412718946949415029080194980493978605678414396606766712262274235284928962561 * 22278 + 21774293793439586643674306888881718167342014062406478752847391700510857054773 + d, d = 0, 4, 6, 10, 12, 16 (766 digits, 1/14/2018, Riecoin #814032)
29696978890366869883141509418765838581871522982358338407613039711378021084519043 * 22259 + 24152316155470595374357736963765392505702343434016117070743766886456802014213 + d, d = 0, 4, 6, 10, 12, 16 (760 digits, 12/31/2017, Riecoin #805968)
29691575669072177222494655186416928710256802541243921484227880404600991044790342 * 22259 + 22953847913844494543791161053509719129919186139904030102712344430311343318911 + d, d = 0, 4, 6, 10, 12, 16 (760 digits, 12/16/2017, Riecoin #797904)
29738370152765841200477916368997470863233149039979929714395166089470825913521999 * 22250 + 3267273123746637724423731592929240166353975680818870504129389950929427468581 + d, d = 0, 4, 6, 10, 12, 16 (757 digits, 2/11/2018, Riecoin #830160)
113225039190926127209 * 2339# / 57057 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20 (1002 digits, 27 Jan 2021, Peter Kaiser)
32821868878860201045633341031688415601401701228 32878265333984717524446848642006351778066196724473 92249620201536539259942023218972369026762290403609 01005487309186655777663859063397693729163631275766 07799875309038457637116938538279395260265064447747 74261236889041020217108597484837589978261046949778 71991825164994665583879769659044973939714534960362 41885200541893611077817261813672809971503287259089 * 317# + 1068701 + d, d = 0, 2, 6, 8, 12, 18, 20 (527 digits, 16 Jun 2019, Vidar Nakling, RIEMINER0.9, PRIMO)
115828580393941*1200# + 5132201 + d, d = 0, 2, 6, 8, 12, 18, 20 (515 digits, 18 Jan 2018, Norman Luhn, PRIMO)
4733578067069 * 940# + 626609 + d, d = 0, 2, 8, 12, 14, 18, 20 (402 digits, May 2016, Norman Luhn)
687001431518312990252195799540952 * 719# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20 (331 digits, 25 Sep 2020, Michalis Christou, Rieminer)
686636073174158279347746711902518 * 719# + 701889794782061 + d, d = 0, 2, 6, 8, 12, 18, 20 (331 digits, 25 Sep 2020, Michalis Christou, Rieminer)
686488342697495738978150794512038 * 719# + 1277156391416021 + d, d = 0, 2, 6, 8, 12, 18, 20 (331 digits, 25 Sep 2020, Michalis Christou, Rieminer)
686305940768787196771563962517233 * 719# + 1487854607298791 + d, d = 0, 2, 6, 8, 12, 18, 20 (331 digits, 25 Sep 2020, Michalis Christou, Rieminer)
686129792256610907998640667932122 * 719# + 701889794782061 + d, d = 0, 2, 6, 8, 12, 18, 20 (331 digits, 25 Sep 2020, Michalis Christou, Rieminer)
685817639451814894948541841801955 * 719# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20 (331 digits, 25 Sep 2020, Michalis Christou, Rieminer)
6879356578124627875380298699944709053335 * 677# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (324 digits, 12 Mar 2021, Michalis Christou)
6492845263546348546 * 700# + 226449521 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (309 digits, 19 Oct 2019, Thomas Nguyen, RIEMINER 0.91)
359378518392551 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)
5602666 54113573653807256039421159095716686744230887487823 55514929545904535010555909219571677274687952903198 * 431# + 1418575498583 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 (281 digits, 29 May 2021, PrimaPool)
49213 11217401936811103526156352113727870867374033023319 46846916505063641420547506943834266263728517203001 * 431# + 235520915372201 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (279 digits, 29 May 2021, ric1qdekpf4v3lwhfu2apasq4metfuuu35q4keu5prq)
1727317194 68357534695186802566959090837823806487250092096759 10868709024745855633473508971497256694964580562010 * 419# + 226374233346623 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 (279 digits, 29 May 2021, PrimaPool)
7972898380320551572902687408841330128076444 69800381838535408958174313617814919077117774769636 * 449# + 114189340938131 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (278 digits, 25 May 2021, Pttn)
190691452886 60724075207124034559127904455846734774801103911151 31600178350904532358406293408041601503459947335601 * 409# + 226554621544613 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 (278 digits, 31 May 2021, PrimaPool)
336156 34685209502103299496432743672107320096495787519596 55486658770545087397881480698253776672526729812074 * 421# + 114023297140211 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (277 digits, 25 May 2021, PrimaPool)
43414128754 21363078855220435892914409488271998218360408303523 02979909931330843256806773392894057481877865551107 * 409# + 226554621544613 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 (277 digits, 29 May 2021, XpoolX)
171261 40574427698443038292751799032279392120235175186111 33550038875063021572377776139492086991109134537768 * 421# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (277 digits, 29 May 2021, XpoolX)
66160221840972226140223807708905566890407723048320 78213536970036538738519759031878588550223823046887 * 409# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (267 digits, 22 May 2021, XpoolX)
60 06559904350134849252155379950254909378835186594558 86375786941368048031982994793132628369636851519027 * 401# + 701889794782061 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (266 digits, 01 May 2021, Suprnova)
218 35722977202308313590097241777421011879112635152949 76358043264898170326365943244347564793366086140266 * 397# + 1277156391416021 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (264 digits, 17 May 2021, PrimaPool)
3483 87853121458649008179383696133766318175683929635059 02067913510453141649847518582456424344347357754742 * 383# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (260 digits, 23 Apr 2021, ric1qmstmv36kuthe7xu4dxzhz6jwnk27t4xqtvmqfa)
244819757603183304556386786638379528596212689 91918113342032150939302877507639213478752614572807 * 401# + 226193845148629 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 (258 digits, 09 May 2021, Pttn)
6309 21912327808796744952939633997746226000817332256293 31799182311085295909231723068059035832331092803319 * 379# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (258 digits, 06 May 2021, XpoolX)
3050506767208880140795226836760067306733856274919 79965295066581348929079873313258875248946926733351 * 383# + 380284918609481 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (255 digits, 06 May 2021, PrimaPool)
69638 39794595034657927780534260078976037788331435278665 70574855127452541573990000208963455191492119007079 * 359# + 701889794782061 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (251 digits, 16 Apr 2021, TomB)
2215574667032155520 01687324835590414397820789810764345732668494831717 17294207993515336980845763232977257432862782607440 * 313# + 855709 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 (247 digits, 10 Apr 2021, ric1quau6a3z8qu4ar204pwgz2vdndyta455vsn99lq)
33521646378383216495527 * 331# + 4700094892301 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (156 digits, 4 Apr 2020, Thomas Nguyen, RIEMINER 0.91, MPZ-APRCL)
772556746441918 * 300# + 29247917 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 (136 digits, 9 Feb 2017, Norman Luhn)
7425 * 281# + 471487291717627721 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (120 digits, May 2016, Roger Thompson)
118557188915212 * 260# + 25658441 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (118 digits, Jun 2014, Norman Luhn)
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 53586844409797545 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (108 digits, 23 Sep 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 51143234991402697 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (108 digits, 23 Sep 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 50679161987995696 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (108 digits, 23 Sep 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 49561325184911775 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (108 digits, 23 Sep 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 49376500222690335 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (108 digits, 23 Sep 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 48866957363924465 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (108 digits, 23 Sep 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 49376500222690335 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (108 digits, 23 Sep 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 46622982649030457 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (108 digits, 23 Sep 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 30796489110940369 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (108 digits, 23 Sep 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 27407861785763183 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (108 digits, 23 Sep 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 20731977215353082 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (108 digits, 23 Sep 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 20118509988610513 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (108 digits, 23 Sep 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 15866045335517629 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (108 digits, 23 Sep 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 5238271627884665 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (107 digits, 23 Sep 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 4471872451082759 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (107 digits, 28 May 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 1296173254392493 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (107 digits, 23 Sep 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 27407861785763183 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (108 digits, 23 Sep 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)
613176722801194*151# + 177321217 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (75 digits, Sep 2014, Michael Stocker, PRIMO)
467756 * 151# + 193828829641176461 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (66 digits, May 2014, Roger Thompson)
59125383480754 * 113# + 12455557957 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (61 digits, Sep 2013, Michael Stocker)
78989413043158 * 109# + 38458151 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (59 digits, Jan 2010, Norman Luhn)
450725899 * 113# + 1748520218561 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (56 digits, Nov 2014, Martin Raab)
14815550 * 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (50 digits, Feb 2013, Roger Thompson)
8486221 * 107# + 4549290807806861 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (50 digits, May 2006, Dirk Augustin & Jens Kruse Andersen)
10000000000000000000000000000929532973818094710897 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (50 digits, 24 Feb 2021, Norman Luhn)
10000000000000000000000000000896396147387349765031 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (50 digits, 24 Feb 2021, Norman Luhn)
4135997219394611 * 110# + 117092849 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (61 digits, 23 Mar 2017, Norman Luhn)
14815550 * 107# + 4385574275277311 + d, d = 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (50 digits, Feb 2013, Roger Thompson)
14815550 * 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (50 digits, Feb 2013, Roger Thompson)
61571 * 107# + 4803194122972361 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (48 digits, Aug 2009, Jens Kruse Andersen)
381955327397348*80# + 18393209 + d, d = 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (46 digits, Dec 2007, Norman Luhn)
381955327397348*80# + 18393209 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (46 digits, Dec 2007, Norman Luhn)
1000000000000000002713562652524314606953 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48 (40 digits, 10 Mar 2021, Norman Luhn)
1000000000000000002334523699629280598673 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36, 46, 48 (40 digits, 10 Mar 2021, Norman Luhn)
1000000000000000000368816080526066037739 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (40 digits, 10 Mar 2021, Norman Luhn)
1000000000000000000349508508460276218891 + d, d = 0, 6, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 48 (40 digits, 10 Mar 2021, Norman Luhn)
14815550 * 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (50 digits, Feb 2013, Roger Thompson)
381955327397348*80# + 18393209 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (46 digits, Dec 2007, Norman Luhn)
1000000000000000000349508508460276218889 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (40 digits, 10 Mar 2021, Norman Luhn)
10000000000009283441665311798539399 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (35 digits, Feb 2021, Norman Luhn)
10000000000001275924044876917671361 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (35 digits, Feb 2021, Norman Luhn)
26093748 * 67# + 383123187762431 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Feb 2005, Christ van Willegen & Jens Kruse Andersen)
108804167016152508211944400342691 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)
107173714602413868775303366934621 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)
101885197790002105359911556070661 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)
101803109763079694387921584406441 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)
33554294028531569*61# + 57800747 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 (40 digits, 25 Jan 2017, Norman Luhn)
322255 * 73# + 1354238543317302647 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 (35 digits, 18 Nov 2016, Roger Thompson)
10004646546202610858599716515809907 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 (35 digits, Sep 2012, Roger Thompson)
302458608131364933637125192102583 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (33 digits, Feb 2021, Roger Thompson)
150048143328514263089612453401301 + d, d = 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62 (33 digits, Feb 2021, Roger Thompson)
107173714602413868775303366934621 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 (33 digits, Apr 2008, Jens Kruse Andersen)
99999999948164978600250563546400 + d, d = 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67 (32 digits, Nov 2004, Joerg Waldvogel and Peter Leikauf)
1251030012595955901312188450381 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 (31 digits, Oct 2003, Hans Rosenthal & Jens Kruse Andersen)
1100916249233879857334075234831 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 (31 digits, Oct 2003, Hans Rosenthal & Jens Kruse Andersen)
1003234871202624616703163933853 + d, d = 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (31 digits, Aug 2012, Roger Thompson)
322255 * 73# + 1354238543317302647 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 (35 digits, 18 Nov 2016, Roger Thompson)
302458608131364933637125192102583 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (33 digits, Feb 2021, Roger Thompson)
150048143328514263089612453401301 + d, d = 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 (33 digits, Feb 2021, Roger Thompson)
1003234871202624616703163933853 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (31 digits, Aug 2012, Roger Thompson)
11413975438568556104209245223 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (29 digits, Jan 2012, Roger Thompson)
5867208169546174917450987997 + d, d = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)
5621078036155517013724659007 + d, d = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)
4668263977931056970475231217 + d, d = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)
4652363394518920290108071167 + d, d = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)
4483200447126419500533043987 + d, d = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)
150048143328514263089612453401301 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 (33 digits, Feb 2021, Roger Thompson)
100845391935878564991556707107 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66 (30 digits, Feb 2013, Roger Thompson)
11413975438568556104209245223 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66 (29 digits, Jan 2012, Roger Thompson)
11410793439953412180643704677 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66 (29 digits, Jan 2012, Roger Thompson)
5867208169546174917450987997 + d, d = 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)
5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)
5621078036155517013724659007 + d, d = 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)
5621078036155517013724659007 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)
4668263977931056970475231217 + d, d = 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)
4668263977931056970475231217 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)
5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)
5621078036155517013724659007 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)
4668263977931056970475231217 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)
4652363394518920290108071167 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)
4483200447126419500533043987 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)
3361885098594416802447362317 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jul 2013, Raanan Chermoni & Jaroslaw Wroblewski)
3261917553005305074228431077 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jul 2013, Raanan Chermoni & Jaroslaw Wroblewski)
3176488693054534709318830357 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jul 2013, Raanan Chermoni & Jaroslaw Wroblewski)
2650778861583720495199114537 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Feb 2013, Raanan Chermoni & Jaroslaw Wroblewski)
2406179998282157386567481191 + d, d = 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 (28 digits, Dec 2012, Raanan Chermoni & Jaroslaw Wroblewski)
622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76 (30 digits, December 27, 2018, Raanan Chermoni & Jaroslaw Wroblewski)
248283957683772055928836513589 + d, d = 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (30 digits, 1 Aug 2016, Raanan Chermoni & Jaroslaw Wroblewski)
138433730977092118055599751669 + d, d = 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (30 digits, 8 Oct 2015, Raanan Chermoni & Jaroslaw Wroblewski)
39433867730216371575457664399 + d, d = 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (29 digits, 8 Jan 2015, Raanan Chermoni & Jaroslaw Wroblewski)
2406179998282157386567481191 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 (28 digits, Dec 2012, Raanan Chermoni & Jaroslaw Wroblewski)
2348190884512663974906615481 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 (28 digits, Dec 2012, Raanan Chermoni & Jaroslaw Wroblewski)
917810189564189435979968491 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 (27 digits, May 2011, Raanan Chermoni & Jaroslaw Wroblewski)
656632460108426841186109951 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 (27 digits, 19 Feb 2011, Raanan Chermoni & Jaroslaw Wroblewski)
630134041802574490482213901 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 (27 digits, 9 Feb 2011, Raanan Chermoni & Jaroslaw Wroblewski)
{37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113}
1236637204227022808686214288579 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (31 digits, May 23, 2021, Raanan Chermoni & Jaroslaw Wroblewski)
1188350591359110800209379560799 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (31 digits, January 21, 2021, Raanan Chermoni & Jaroslaw Wroblewski)
1153897621507935436463788957529 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (31 digits, December 26, 2020, Raanan Chermoni & Jaroslaw Wroblewski)
1135540756371356698957890225821 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (31 digits, December 19, 2020, Raanan Chermoni & Jaroslaw Wroblewski)
1126002593922465663847897293731 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (31 digits, November 17, 2020, Raanan Chermoni & Jaroslaw Wroblewski)
1094372814043722195189448411199 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (31 digits, October 20, 2020, Raanan Chermoni & Jaroslaw Wroblewski)
1060475118776959297139870952701 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (31 digits, September 18, 2020, Raanan Chermoni & Jaroslaw Wroblewski)
999627565307688186459783232931 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (30 digits, June 19, 2020, Raanan Chermoni & Jaroslaw Wroblewski)
957278727962618711849051282459 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, March 23, 2020, Raanan Chermoni & Jaroslaw Wroblewski)
839013472011818416634745523991 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (30 digits, October 28, 2019, Raanan Chermoni & Jaroslaw Wroblewski)
622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76, 82, 84 (30 digits, December 27, 2018, Raanan Chermoni & Jaroslaw Wroblewski)
248283957683772055928836513589 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (30 digits, 1 Aug 2016, Raanan Chermoni & Jaroslaw Wroblewski)
138433730977092118055599751669 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (30 digits, 8 Oct 2015, Raanan Chermoni & Jaroslaw Wroblewski)
39433867730216371575457664399 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (29 digits, 8 Jan 2015, Raanan Chermoni & Jaroslaw Wroblewski)
{29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113}
| The largest known prime k-tuplets | ||||
| k | Digits | Prime k-tuplet | Who | When |
| 1 | 24862048 | 282589933 − 1 | P. Laroche, G. Woltman, S. Kurowski, A. Blosser, et al (GIMPS) | 21 Dec 2018 |
| 2 | 388342 | 2996863034895 * 21290000 ± 1 | Tom Greer, TWINGEN, PRIMEGRID, LLR | Sep 2016 |
| 3 | 20008 | 4111286921397 * 266420 + d, d = −1, 1, 5 | Peter Kaiser, POLYSIEVE, LLR, PRIMO | 24 Apr 2019 |
| 4 | 10132 | 667674063382677 * 233608 + d, d = −1, 1, 5, 7 | Peter Kaiser, PRIMO | 27 Feb 2019 |
| 5 | 2034 | 566761969187 * 4733#/2 + d, d = −8, −4, −2, 2, 4 | Serge Batalov, NEWPGEN, OPENPFGW, PRIMO | December 2020 |
| 6 | 1037 | 28993093368077 * 2400# + 19417 + d, d = 0, 4, 6, 10, 12, 16 | Norman Luhn, APSIEVE, PRIMO | 14 Mar 2016 |
| 7 | 1002 | 113225039190926127209 * 2339# / 57057 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20 | Peter Kaiser | 27 Jan 2021 |
| 8 | 324 | 6879356578124627875380298699944709053335 * 677# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 | Michalis Christou, RIEMINER 0.91 | 12 Mar 2021 |
| 9 | (277 | 171261 40574427698443038292751799032279392120235175186111 33550038875063021572377776139492086991109134537768 * 421# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 | XpoolX | 29 May 2021 |
| 10 | 156 | 33521646378383216495527 * 331# + 4700094892301 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 | Thomas Nguyen, RIEMINER 0.91, MPZ-APRCL | 4 Apr 2020 |
| 11 | 108 | 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 49376500222690335 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 | Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO | 23 Sep 2019 |
| 12 | 108 | 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 27407861785763183 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 | Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO | 23 Sep 2019 |
| 13 | 61 | 4135997219394611 * 110# + 117092849 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 | Norman Luhn | 23 Mar 2017 |
| 14 | 50 | 14815550 * 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 | Roger Thompson | Feb 2013 |
| 15 | 40 | 33554294028531569*61# + 57800747 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 | Norman Luhn | 25 Jan 2017 |
| 16 | 35 | 322255 * 73# + 1354238543317302647 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 | Roger Thompson | 18 Nov 2016 |
| 17 | 33 | 150048143328514263089612453401301 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 | Roger Thompson | Feb 2021 |
| 18 | 28 | 5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 | Raanan Chermoni & Jaroslaw Wroblewski | Mar 2014 |
| 19 | 30 | 622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76 | Raanan Chermoni & Jaroslaw Wroblewski | 27 Dec 2018 |
| 20 | 31 | 1236637204227022808686214288579 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 | Raanan Chermoni & Jaroslaw Wroblewski | May 23, 2021 |
| 21 | 30 | 622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76, 82, 84 | Raanan Chermoni & Jaroslaw Wroblewski | 27 Dec 2018 |
| 22 | 2 | {23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113} | - | - |
| 23 | 2 | {19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113} | - | - |
| 24 | - | There are no known prime 24-tuplets | - | - |
| First appearance of a non-trivial prime k-tuplet | ||||
| k | Digits | Prime k-tuplet | Who | When |
| <12 | - | No reliable information | - | - |
| 12 | 13 | 1418575498567 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 | D. Betsis & S. Säfholm | 1982 |
| 13 | 14 | 10527733922579 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 | D. Betsis & S. Säfholm | 1982 |
| 14 | 17 | 21817283854511261 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 | D. Betsis & S. Säfholm | 1982 |
| 15 | 21 | 347709450746519734877 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 | Joerg Waldvogel | 1996 |
| 16 | 21 | 347709450746519734877 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 | Joerg Waldvogel | 1996 |
| 17 | 22 | 1620784518619319025971 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 | Joerg Waldvogel | 1997 |
| 18 | 25 | 2845372542509911868266807 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 | Joerg Waldvogel & Peter Leikauf | 14 Nov 2000 |
| 19 | 27 | 630134041802574490482213901 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 | Raanan Chermoni & Jaroslaw Wroblewski | 9 Feb 2011 |
| 20 | 28 | 3941119827895253385301920029 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 | Raanan Chermoni & Jaroslaw Wroblewski | 24 Jun 2014 |
| 21 | 29 | 39433867730216371575457664399 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 | Raanan Chermoni & Jaroslaw Wroblewski | 8 Jan 2015 |
| First appearance of 100 digits | ||||
| k | Digits | Prime k-tuplet | Who | When |
| 1 | 157 | 2521 − 1 | R. M. Robinson | Jan 1952 |
| 2-5 | - | No reliable information | - | - |
| 6 | 133 | 2 * 10132 + 75543532187 + d, d = 0, 4, 6, 10, 12, 16 | Tony Forbes | Apr 1994 |
| 7 | 104 | 4293326603 * 233# + 399389 + d, d = 0, 2, 8, 12, 14, 18, 20 | Radoslaw Naleczynski | Dec 1998 |
| 8 | 110 | 388793398651 * 250# + 1042090781 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 | Norman Luhn | Feb 2001 |
| 9 | 110 | 388793398651 * 250# + 1042090781 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 | Norman Luhn | Feb 2001 |
| 10 | 103 | 72613488698235 * 227# + 39058751 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 | Norman Luhn | Apr 2004 |
| 11 | 104 | 24698258 * 239# + 28606476153371 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 | Norman Luhn & Jens Kruse Andersen | Aug 2004 |
| 12 | 108 | 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 27407861785763183 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 | Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO | 23 Sep 2019 |
| First appearance of 1000 digits | ||||
| k | Digits | Prime k-tuplet | Who | When |
| 1 | 1332 | 24423 − 1 | Alexander Hurwitz | Nov 1961 |
| 2 | 1040 | 256200945 * 23426 ± 1 | Oliver Atkin & N. W. Rickert | 1980 |
| 3 | 1083 | 437850590*(23567 − 21189) − 6*21189 + d, d = −5, −1, 1 | Tony Forbes | Dec 1996 |
| 4 | 1004 | 76912895956636885*(23279 − 21093) − 6*21093 + d, d = −7, −5, −1, 1 | Tony Forbes | Sep 1998 |
| 5 | 1034 | 31969211688*2400# + 16061 + d, d = 0, 2, 6, 8, 12 | Norman Luhn | Jul 2002 |
| 6 | 1037 | 28993093368077 * 2400# + 19417 + d, d = 0, 4, 6, 10, 12, 16 | Norman Luhn, APSIEVE, PRIMO | 14 Mar 2016 |
| 7 | 1002 | 113225039190926127209 * 2339# / 57057 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20 | Peter Kaiser | 27 Jan 2021 |
| First appearance of 10000 digits | ||||
| k | Digits | Prime k-tuplet | Who | When |
| 1 | 13395 | 244497 − 1 | Harry Nelson & David Slowinski | Apr 1979 |
| 2 | 11713 | 242206083 * 238880 ± 1 | H. K. Indlekofer & A. Járai | Nov 1995 |
| 3 | 10047 | 2072644824759 * 233333 + d, d = −1, 1, 5 | Norman Luhn, François Morain, FastECPP | Nov 2008 |
| 4 | 10132 | 667674063382677 * 233608 + d, d = −1, 1, 5, 7 | Peter Kaiser, PRIMO | 27 Feb 2019 |
| First appearance of 100000 digits | ||||
| k | Digits | Prime k-tuplet | Who | When |
| 1 | 227832 | 2756839 − 1 | David Slowinski & Paul Gage | Apr 1992 |
| 2 | 100355 | 65516468355 * 2333333 ± 1 | Peter Kaiser, NEWGEN, PRIMEGRID, TPS, LLR | Aug 2009 |
| First appearance of 1000000 digits | ||||
| k | Digits | Prime k-tuplet | Who | When |
| 1 | 2098960 | 26972593 − 1 | Nayan Hajratwala, George Woltman, Scott Kurowski et al (GIMPS) | Jun 1999 |
| First appearance of 10000000 digits | ||||
| k | Digits | Prime k-tuplet | Who | When |
| 1 | 12978189 | 243112609 − 1 | Edson Smith, George Woltman, Scott Kurowski et al (GIMPS) | Sep 2008 |
List of all possible patterns of prime k-tuplets
List of the smallest prime k-tuplets
Various lists of prime k-tuplets
Near misses: Clusters of primes that didn't quite make it into the main list
The Hardy-Littlewood constants pertaining to the distribution of prime k-tuplets [HL22]
Jens Kruse Andersen: The Largest Known Simultaneous Primes
Jens Kruse Andersen: Consecutive Primes in Arithmetic Progression
Jens Kruse Andersen: Largest Consecutive Factorizations
Dirk Augustin: Cunningham Chain records
Chris K. Caldwell: The Largest Known Primes
Chris K. Caldwell: Top twenty twin primes
TF: Ten consecutive primes in arithmetic progression
Dr. Minh. L. Perez Press: Smarandache Primes
N. J. A. Sloane: On-Line Encyclopedia of Integer Sequences
Manfred Toplic: The Nine and Ten Primes Project
Robin Whitty: Theorem of the Day
A prime k-tuplet is a sequence of k consecutive prime numbers such that in some sense the difference between the first and the last is as small as possible. The idea is to generalise the concept of prime twins.
More precisely: We first define s(k) to be the smallest number s for which there exist k integers b1 < b2 < ... < bk, bk − b1 = s and, for every prime q, not all the residues modulo q are represented by b1, b2, ..., bk. A prime k-tuplet is then defined as a sequence of consecutive primes {p1, p2, ..., pk} such that for every prime q, not all the residues modulo q are represented by p1, p2, ..., pk, pk − p1 = s(k). Observe that the definition might exclude a finite number (for each k) of dense clusters at the beginning of the prime number sequence - for example, {97, 101, 103, 107, 109} satisfies the conditions of the definition of a prime 5-tuplet , but {3, 5, 7, 11, 13} doesn't because all three residues modulo 3 are represented.
The simplest case is s(2) = 2, corresponding to prime twins: {p, p + 2}. Next, s(3) = 6 and two types of prime triplets: {p, p + 2, p + 6} and {p, p + 4, p + 6}, followed by s(4) = 8 with just one pattern: {p, p + 2, p + 6, p + 8} of prime quadruplets. The sequence continues with s(5) = 12, s(6) = 16, s(7) = 20, s(8) = 26, s(9) = 30, s(10) = 32, s(11) = 36, s(12) = 42, s(13) = 48, s(14) = 50, s(15) = 56, s(16) = 60, s(17) = 66 and so on. It is number A008407 in N.J.A. Sloane's On-line Encyclopedia of Integer Sequences.
In keeping with similar published lists, I have decided not to accept anything other than true, proven primes. Numbers which have merely passed the Fermat test, aN = a (mod N), will need to be validated. If N − 1 or N + 1 is sufficiently factorized (usually just under a third), the methods of Brillhart, Lehmer and Selfridge [BLS75] will suffice. Otherwise the numbers may have to be subjected to a general primality test, such as the Jacobi sum test of Adleman, Pomerance, Rumely, Cohen and Lenstra (APRT-CLE in UBASIC, for example), or one of the elliptic curve primality proving programs: Atkin and Morain's ECPP, or its successor, Franke, Kleinjung, Wirth and Morain's FAST-ECPP, or Marcel Martin's PRIMO.
Euclid proved that there are infinitely many primes. Paulo Ribenboim [Rib95] has collected together a considerable number of different proofs of this important theorem. My favourite (which is not in Ribenboim's book) goes like this: We have
∏p prime 1/(1 − 1/p2) = ∑n = 1 to ∞ 1/n2 = π2/6.
But π2 is irrational; so the product on the left cannot have a finite number of factors.
In its simplest form, the prime number theorem states that the number of primes less than x is asymptotic to x/(log x). This was first proved by Hadamard and independently by de la Vallee Poussin in 1896. Later, de la Vallee Poussin found a better estimate:
∫u = 0 to x du/(log u) + error term,
where the error term is bounded above by A x exp(−B √(log x)) for some constants A and B. With more work (H.-E. Richert, 1967), √(log x) in this last expression can be replaced by (log x)3/5(log log x)−1/5. The most important unsolved conjecture of prime number theory, indeed, all of mathematics - the Riemann Hypothesis - asserts that the error term can be bounded by a function of the form A √x log x.
G.H. Hardy & J.E. Littlewood did the first serious work on the distribution of prime twins. In their paper 'Some problems of Partitio Numerorum: III...' [HL22], they conjectured a formula for the number of twins between 1 and x:
2 C2 x / (log x)2,
where C2 = ∏p prime, p > 2 p(p − 2) / (p − 1)2 = 0.66016... is known as the twin prime constant.
V. Brun showed that the sequence of twins is thin enough for the series ∑p and p + 2 prime 1 / p to converge. The twin prime conjecture states that the sum has infinitely many terms. The nearest to proving the conjecture is Jing-Run Chen's result that there are infinitely many primes p such that p + 2 is either prime or the product of two primes [HR73].
The Partitio Numerorum: III paper [HL22] goes on to formulate a general conjecture concerning the distribution of arbitrary groups of prime numbers (The k-tuplets of this site are special cases): Let b1, b2, ..., bk be k distinct integers. Then the number of groups of primes N + b1, N + b2, ..., N + bk between 2 and x is approximately
Hk Ck ∫u = 2 to x du / (log u)k,
where
Hk = ∏p prime, p ≤ k pk − 1 (p − v) / (p − 1)k ∏p prime, p > k, p|D (p − v) / (p − k),
Ck = ∏p prime, p > k pk − 1 (p − k) / (p − 1)k,
v = v(p) is the number of distinct remainders of b1, b2, ..., bk modulo p and D is the product of the differences |bi − bj|, 1 ≤ i < j ≤ k.
The first product in Hk is over the primes not greater than k, the second is over the primes greater than k which divide D and the product Ck is over all primes greater than k. If you put k = 2, b1 = 0 and b2 = 2, then v(2) = 1, v(p) = p − 1 for p > 2, H2 = 2, and Ck = C2, the twin prime constant given above.
It is worth pointing out that with modern mathematical software the prime k-tuplet constants Ck can be determined to great accuracy. The way not to do it is to use the defining formula. Unless you are very patient, calculating the product over a sufficient number of primes for, say, 20 decimal place accuracy would not be feasible. Instead there is a useful transformation originating from the product formula for the Riemann ζ function:
log Ck = − ∑n = 2 to ∞ log [ζ(n) ∏p prime, p ≤ k (1 − 1/pn)] / n ∑d|n μ(n/d) (kd − k).
[BLS75] John Brillhart, D.H. Lehmer & J.L. Selfridge, New primality criteria and factorizations of 2m ± 1, Math. Comp. 29 (1975), 620-647.
[CD93] C.K.Caldwell & H. Dubner, Primorial, factorial and multifactorial primes, Math. Spectrum 26 (1993/94), 1-7.
[F97f] Tony Forbes, Prime 17-tuplet, NMBRTHRY Mailing List, September 1997.
[F02] Tony Forbes, Titanic prime quintuplets, M500 189 (December, 2002), 12-13.
[F09] Tony Forbes, Gigantic prime triplets, M500 226 (February, 2009), 18-19.
[Guy94] Richard K. Guy, Unsolved Problems in Number Theory, second edn., Springer-Verlag, New York 1994.
[HL22] G. H. Hardy and J. E. Littlewood, Some problems of Partitio Numerorum: III; on the expression of a number as a sum of primes, Acta Mathematica 44 (1922), 1-70.
[HR73] H. Halberstam and H.-E Richert, Sieve Methods, Academic Press, London 1973.
[Rib95] P. Ribenboim, The New Book of Prime Number Records, 3rd edn., Springer-Verlag, New York 1995
[R95] Warut Roonguthai, Prime quadruplets, NMBRTHRY Mailing List, September 1995.
[R96a] Warut Roonguthai, Prime quadruplets, M500 148 (February 1996), 9.
[R96b] Warut Roonguthai, Large prime quadruplets, NMBRTHRY Mailing List, September 1996.
[R96c] Warut Roonguthai, Large prime quadruplets, M500, 153 (December, 1996), 4-5.
[R97a] Warut Roonguthai, Large prime quadruplets, NMBRTHRY Mailing List, September 1997.
[R97b] Warut Roonguthai, Large prime quadruplets, M500 158 (November 1997), 15.
[S95] G. John Stevens, Prime Quadruplets, J. Recr. Math. 27 (1995), 17-22.