Smallest Prime k-tuplets Oliver Atkin suggested that it would be of interest to record the smallest prime k-tuplet for each type of pattern. In many cases it is trivial; for example, the two types of triplets are {5, 7, 11} and {7, 11, 13}. (We exclude {3, 5, 7} because here all three residues modulo 3 are represented.) With larger k it is not so easy. For instance, the smallest 20-tuplet of the pattern {0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80} is {29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109}. However, there are no known examples of the mirror-image pattern {0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80} Contributions to this section are welcome. k=2 s=2 B={0 2} 1 3 5 2 5 7 3 11 13 4 17 19 5 29 31 6 41 43 7 59 61 8 71 73 9 101 103 10 107 109 k=3 s=6 B={0 4 6} 1 7 11 13 2 13 17 19 3 37 41 43 4 67 71 73 5 97 101 103 6 103 107 109 7 193 197 199 8 223 227 229 9 277 281 283 10 307 311 313 k=3 s=6 B={0 2 6} 1 5 7 11 2 11 13 17 3 17 19 23 4 41 43 47 5 101 103 107 6 107 109 113 7 191 193 197 8 227 229 233 9 311 313 317 10 347 349 353 k=4 s=8 B={0 2 6 8} 1 5 7 11 13 2 11 13 17 19 3 101 103 107 109 4 191 193 197 199 5 821 823 827 829 6 1481 1483 1487 1489 7 1871 1873 1877 1879 8 2081 2083 2087 2089 9 3251 3253 3257 3259 10 3461 3463 3467 3469 k=5 s=12 B={0 4 6 10 12} 1 7 11 13 17 19 2 97 101 103 107 109 3 1867 1871 1873 1877 1879 4 3457 3461 3463 3467 3469 5 5647 5651 5653 5657 5659 6 15727 15731 15733 15737 15739 7 16057 16061 16063 16067 16069 8 19417 19421 19423 19427 19429 9 43777 43781 43783 43787 43789 10 79687 79691 79693 79697 79699 k=5 s=12 B={0 2 6 8 12} 1 5 7 11 13 17 2 11 13 17 19 23 3 101 103 107 109 113 4 1481 1483 1487 1489 1493 5 16061 16063 16067 16069 16073 6 19421 19423 19427 19429 19433 7 21011 21013 21017 21019 21023 8 22271 22273 22277 22279 22283 9 43781 43783 43787 43789 43793 10 55331 55333 55337 55339 55343 k=6 s=16 B={0 4 6 10 12 16} 1 7 11 13 17 19 23 2 97 101 103 107 109 113 3 16057 16061 16063 16067 16069 16073 4 19417 19421 19423 19427 19429 19433 5 43777 43781 43783 43787 43789 43793 6 1091257 1091261 1091263 1091267 1091269 1091273 7 1615837 1615841 1615843 1615847 1615849 1615853 8 1954357 1954361 1954363 1954367 1954369 1954373 9 2822707 2822711 2822713 2822717 2822719 2822723 10 2839927 2839931 2839933 2839937 2839939 2839943 k=7 s=20 B={0 2 6 8 12 18 20} 1 11 13 17 19 23 29 31 2 165701 165703 165707 165709 165713 165719 165721 3 1068701 1068703 1068707 1068709 1068713 1068719 1068721 4 11900501 11900503 11900507 11900509 11900513 11900519 11900521 5 15760091 15760093 15760097 15760099 15760103 15760109 15760111 6 18504371 18504373 18504377 18504379 18504383 18504389 18504391 7 21036131 21036133 21036137 21036139 21036143 21036149 21036151 8 25658441 25658443 25658447 25658449 25658453 25658459 25658461 9 39431921 39431923 39431927 39431929 39431933 39431939 39431941 10 45002591 45002593 45002597 45002599 45002603 45002609 45002611 k=7 s=20 B={0 2 8 12 14 18 20} 1 5639 5641 5647 5651 5653 5657 5659 2 88799 88801 88807 88811 88813 88817 88819 3 284729 284731 284737 284741 284743 284747 284749 4 626609 626611 626617 626621 626623 626627 626629 5 855719 855721 855727 855731 855733 855737 855739 6 1146779 1146781 1146787 1146791 1146793 1146797 1146799 7 6560999 6561001 6561007 6561011 6561013 6561017 6561019 8 7540439 7540441 7540447 7540451 7540453 7540457 7540459 9 8573429 8573431 8573437 8573441 8573443 8573447 8573449 10 17843459 17843461 17843467 17843471 17843473 17843477 17843479 *** Information for k = 8 provided by Gary Barnes k=8 s=26 B={0 2 6 8 12 18 20 26} Initial members of prime 8-tuplets: 11, 15760091, 25658441, 93625991, 182403491, 226449521, 661972301, 910935911 k=8 s=26 B={0 2 6 12 14 20 24 26} Initial members of prime 8-tuplets: 17, 1277, 113147, 2580647, 20737877, 58208387, 73373537, 76170527, 100658627, 134764997, 137943347, 165531257, 171958667, 224008217, 252277007, 294536147, 309740987, 311725847, 364154027, 408936947, 515447747, 521481197, 528272177, 619010297, 626927447, 682809977, 701679047, 989043047 k=8 s=26 B={0 6 8 14 18 20 24 26} Initial members of prime 8-tuplets: 88793, 284723, 855713, 1146773, 6560993, 69156533, 74266253, 218033723, 261672773, 302542763, 964669613 *** Information for k = 9 provided by Vladimir Vlesycit and Matt Andersen k=9 s=30 B={0 4 6 10 16 18 24 28 30} Initial members of prime 9-tuplets: 13, 113143, 626927443, 2335215973, 3447123283, 4086982633, 4422726013, 6318867403, 7093284043, 8541306853, 10998082213, 14005112893, 18869466373, 21528117883, 21843411823, 28156779793, 30303283243, 31194463033, 33081664153, 35004115033, 35193551203, 35193551203, 39682755793, 41666557393, 47064359803, 47798406193, 49533488083, 51435506383, 52719078673, 57366516913, 59038943053, 59780979103, 62564104063, 75832757233, 79963083433, 83122625473, 93138486583, 93789236683, 96259498873, 97966866703, 99409572523, 103050287293, 103798521703, 104751695743, 106211658583, 107961509413, 109055504263, 114829973953, 115316796583, 118423267423, 118934479663, 122015998183, 124684085413, 131971411123, 135936506593, 137064385303, 140163144673, 142641528763, 143449458613, 151411248403, 154176423673, 157797433873, 158114143063, 163680506593, 168478035613, 175029719713, 175956805873, 176615006983, 177083148013, 179822268163, 180386310733, 188652262213, 189540653983, 194666570713, 197176517953, 198871424023, 201622282633, 201686318833, 202500011713, 203626335373, 205750546273, 209853250543, 217636995463, 218018750683, 222030503323, 224372900023, 233121174043, 233691050683, 236345972893 k=9 s=30 B={0 4 10 12 18 22 24 28 30} Initial members of prime 9-tuplets: 88789, 855709, 74266249, 964669609, 1422475909, 2117861719, 2558211559, 2873599429, 5766036949, 6568530949, 8076004609, 9853497739, 16394542249, 21171795079, 21956291869, 22741837819, 26486447149, 27254489389, 36955907689, 37045175329, 39745733539, 71369039869, 75884569279, 87554295529, 88431318949, 108039383509, 120739615669, 123878599699, 141741265639, 146163774919, 164444511589, 177313452289, 179590045489, 181523196289, 185222773639, 187996202479, 188506859059, 193993844899, 194000607319, 196253905009, 199998949669, 216887683069, 217999764109, 219795770629, 231255798859, 242360943259, 260222396869, 267204456499, 292624431079, 293320920019, 303023883289, 330293107579, 341741739319, 355111401499, 360744936679, 385736835919, 413801829799, 437984753179, 448301185459, 463152187009, 465328692529, 478863103699, 489963274039, 533840361139, 557552371279, 609587549089, 614414530969, 618697706299, 619833080539, 627929073829, 642152975899, 662883608899, 666413245009, 672422947099, 696391309699, 709116660379, 716726260489, 725483102779, 740312270209, 782422789849 k=9 s=30 B={0 2 6 8 12 18 20 26 30} Initial members of prime 9-tuplets: 11, 182403491, 226449521, 910935911, 1042090781, 1459270271, 2843348351, 6394117181, 6765896981, 8247812381, 8750853101, 11076719651, 12850665671, 17140322651, 22784826131, 24816950771, 33081664151, 41614070411, 43298074271, 43813839521, 53001578081, 54270148391, 57440594201, 64202502431, 66449431661, 80587471031, 83122625471, 84882447101, 93288681371, 98257943081, 101698684931, 117762315941, 121317512201, 139017478331, 148323246341, 165952761041, 169349651741, 173405293331, 180237252311, 185814366581, 187735172741, 201708760061, 203499800501, 231965242121, 232942479221, 238333854371, 250319013011, 260208736631, 276512899961, 294757438511, 294920291201, 297532090391, 311094184391, 312962494961, 319751660141, 356380654361, 362110374161, 383960791211, 447073236341, 458958087431, 467061716111, 480327521741, 486294251741, 490609410281 k=9 s=30 B={0 2 6 12 14 20 24 26 30} Initial members of prime 9-tuplets: 17, 1277, 113147, 252277007, 408936947, 521481197, 1116452627, 1209950867, 1645175087, 2966003057, 3947480417, 6234613727, 9307040837, 9853497737, 11878692167, 13766391467, 21956291867, 22741837817, 24388061207, 24718113437, 28498194707, 29177275067, 31201358387, 32417170127, 36230354177, 42052236197, 51435506387, 54202818407, 58797275477, 65366478887, 67406579477, 71999669297, 78270669767, 78271296197, 80187554327, 83026673747, 83115690167, 94968471107, 98793034577, 99409572527, 103798521707, 105035735657, 107735906477, 110559120797, 115154953997, 119187533807, 121801892477, 127788668717, 130714028567, 133021481687, 138733641017, 143977593497, 144011113217, 149414528357, 150684085817, 153039812897, 158532152477, 164444511587, 168478035617, 172660174847, 179142250577, 179590045487, 180566297267, 181503415127, 183562654127, 185242953017, 187505538587, 193378900697, 197335238117, 205137898307, 209853250547, 209926512767, 211298118587, 213243428597, 213252021107, 216741880697, 217999764107, 224745399077, 227520127817, 231255798857, 242360943257, 249269868557 *** Information for k = 10 provided by Warut Roonguthai and Joerg Waldvogel k=10 s=32 B={0 2 6 8 12 18 20 26 30 32} 11, 33081664151, 83122625471, 294920291201, 573459229151, 663903555851, 688697679401, 730121110331, 1044815397161, 1089869189021, 1108671297731, 1235039237891, 1291458592421, 1738278660731, 1761428046221, 1769102630411, 1804037746781, 1944541048181, 2135766611591, 2177961410741, 2206701370871, 2395426439291, 2472430065221, 2601621164951, 2690705555381, 2744825921681, 2745579799421, 2772177619481, 2902036214921, 3132826508081, 3447850801511, 3640652828621, 3692145722801, 4136896626671, 4360784021591, 4563595602101, 4582294051871, 4700094892301, 5289415841441, 5308007072981, 5351833972601, 5731733114951, 5912642968691, 5923626073901, 6218504101541, 7353767766461, 7535450701391, 7559909355611, 7908189600581, 9062538296081, 9494258167391, 9898228819091 k=10 s=32 B={0 2 6 12 14 20 24 26 30 32} 9853497737, 21956291867, 22741837817, 164444511587, 179590045487, 217999764107, 231255798857, 242360943257, 666413245007, 696391309697, 867132039857, 974275568237, 976136848847, 1002263588297, 1086344116367, 1403337211247, 1418575498577, 2118274828907, 2202874566017, 2385504217067, 2524133138687, 2935239433097, 3019265734337, 3481009682387, 3692468438357, 3699811791647, 4074379299227, 4258482896177, 4396774576277, 5111078123507, 5248370051777, 5603394155057, 5658223464977, 6302879942057, 6353090848187, 6368171154197, 6390526086797, 6953798916917, 6986122067777, 7117924769927, 7327435302347, 7624513711247, 7679352605357, 8310209124377, 8639103445097, 9378647660507, 9407507858447, 9479280868217, 9506905472147, 9611678132027 *** Information for k = 11 provided by Vladimir Vlesycit and Joerg Waldvogel k=11 s=36 B={0 4 6 10 16 18 24 28 30 34 36} 1418575498573, 2118274828903, 4396774576273, 6368171154193, 6953798916913, 27899359258003, 28138953913303, 34460918582323, 40362095929003, 42023308245613, 44058461657443, 61062361183903, 76075560855373, 80114623697803, 84510447435493, 85160397055813, 90589658803723, 90793299817453, 147119918235523, 156393789335863, 170185081150933, 173469325674493, 176053639140073, 186460616596333, 186938838927073, 195647411389663, 199133130576613, 204757718740423, 218021188549243, 234280497145543, 249826857614413, 282854319391723, 292186774388203, 297416007172363, 317696609400403, 345120905374093, 346117552180633, 352270406993353, 392842699733563, 413564145244483 k=11 s=36 B={0 2 6 8 12 18 20 26 30 32 36} 11, 7908189600581, 10527733922591, 12640876669691, 38545620633251, 43564522846961, 60268613366231, 60596839933361, 71431649320301, 79405799458871, 109319665100531, 153467532929981, 171316998238271, 216585060731771, 254583955361621, 259685796605351, 268349524548221, 290857309443971, 295606138063121, 380284918609481, 437163765888581, 440801576660561, 487925557660811 *** Information for k = 12 provided by Vladimir Vlesycit and Joerg Waldvogel (< 10^16) and Matt Anderson (> 10^16) k=12 s=42 B={0 6 10 12 16 22 24 30 34 36 40 42} 1418575498567, 27899359257997, 34460918582317, 76075560855367, 186460616596327, 218021188549237, 234280497145537, 282854319391717, 345120905374087, 346117552180627, 604439135284057, 727417501795057, 1041814617748747, 1090754719898917, 1539765965257747, 3152045700948217, 3323127757029307, 3449427485143867, 4422879865247917, 4525595253333997, 4730773080017827, 5462875671033007, 6147764065076707, 6205707895751437, 6308411019731047, 7582919852522857, 7791180222409657, 9162887985581557, 9305359177794907, 10096106139749857, 10349085616714687, 10744789916260627, 10932016019429347, 11140102475962687, 12448240792011097, 14727257011031407, 16892267700442207, 17963729763800047, 18908121647739397, 19028992697498857, 19756696515786457, 20252223877980937, 20429666791949257, 21680774776901467, 21682173462980257, 23076668788453507, 24036602580170407, 24101684579532787, 25053289894907347, 25309078073142937, 25662701041982077, 25777719656829367, 26056424604564427, 26315911419972247, 26866456999592437, 26887571851660747, 27303559129791787, 27839080743588187, 28595465291933767, 29137316070747727, 30824439453812077, 31395828815154877, 31979851757518507, 32897714831936797, 33850998835087507, 36147660266252377, 37072866353096647, 37141494251796007, 37489481237373007, 38006810209768627, 38748333093144517, 38994703724306557, 39797843204594317 k=12 s=42 B={0 2 6 8 12 18 20 26 30 32 36 42} 11, 380284918609481, 437163765888581, 701889794782061, 980125031081081, 1277156391416021, 1487854607298791, 1833994713165731, 2115067287743141, 2325810733931801, 3056805353932061, 3252606350489381, 3360877662097841, 3501482688249431, 3595802556731501, 3843547642594391, 5000014653723821, 5861268883004651, 7486645325734691, 7933248530182091, 8760935349271991, 8816939536219931, 8871465225933041, 9354490866900911, 13764730155211151, 13884748604026031, 17438667992681051, 20362378935668501, 20471700514990841, 20475715985020181, 20614750499829371, 21465425387541251, 21628360938574121, 21817283854511261, 22238558064758921, 22318056296221571, 22733842556089781, 22849881428489231, 23382987892499351, 23417442472403711, 25964083184094941, 26515897161980111, 29512383574028471, 30074756036270831, 30310618347929651, 30402250951007051, 30413977411117031, 33502273017038711, 33508988966488151, 33976718302847051, 34783522781262371, 37564605737538611, 37606024583356961, 39138758504100371, 40205947750578341, 40257009922154141, 40392614725338761, 40504121267225501, 41099072498143391, 41289201480321911, 41543933848913381, 42218492028808211, 43938526447515431, 45577046471292221, 46428559244382431, 47009705561193491, 47493758956860101, 48897378456286091, 49242777550551701, 49600456951571411, 49600456951571411, 49719485618652581, 50155365997396391, 50428186330336931, 51553155978279071, 52018707666681641, 57145775215328531, 57853108424841461, 60087392674669091, 60639922253220041, 60948080389385921, 61187849081864621, 62958926374779551 *** Information for k = 13 provided by Vladimir Vlesycit k=13 s=48 B={0 6 12 16 18 22 28 30 36 40 42 46 48} 186460616596321, 7582919852522851, 31979851757518501, 49357906247864281, 79287805466244211, 85276506263432551, 89309633704415191, 89374633724310001, 98147762882334001, 136667406812471371, 137803293675931951, 152004604862224951, 157168285586497021, 159054409963103491, 191495873018073691, 210509800417460611, 301056055256880991, 379374319296406501, 388218745698928981, 389259253111633741 k=13 s=48 B={0 4 6 10 16 18 24 28 30 34 40 46 48} 13, 4289907938811613, 5693002600430263, 21817283854511263, 48290946353555023, 51165618791484133, 53094081535451893, 70219878257874463, 98633358468021313, 99142644093930373, 104814760374339133, 166784569423739203, 167841416726358493, 184601252515266523, 263331429949004353, 272039012072134243, 339094624362619243, 363319822006646623, 363760043662280383, 437335541550793003, 455289126169953193 k=13 s=48 B={0 4 6 10 16 18 24 28 30 34 36 46 48} 1707898733581273, 3266590043460823, 4422879865247923, 10907318641689703, 32472302129057023, 52590359764282573, 60229684381540753, 67893346321234513, 93179596929433093, 115458868925574253, 140563537593599353, 142977538681261363, 148877505784397623, 166362638531783773, 232442516762530153, 236585787518684683, 255933372890105143, 317294052871840123, 325853825645632363, 337188071215909993, 344447962857168403 k=13 s=48 B={0 2 6 8 12 18 20 26 30 32 36 42 48} 11, 7933248530182091, 20475715985020181, 21817283854511261, 33502273017038711, 40257009922154141, 49242777550551701, 49600456951571411, 75093141517810301, 84653373093824651, 119308586807395871, 129037438367672951, 129706953139869221, 151242381725881331, 158947009165390331, 161216594737343261, 167317340088093311, 176587730173540571, 178444395317213141, 197053322268438521, 301854920123441801 k=13 s=48 B={0 2 8 14 18 20 24 30 32 38 42 44 48} 7697168877290909, 10071192314217869, 11987120084474369, 28561589689237439, 62321320746357689, 73698766709402669, 75046774774314359, 79287805466244209, 98551408299919409, 136720189890477209, 225735856757596019, 234065221633327919, 302834818301440259, 360345440708336099, 385443070970192069, 387494664213890249, 466256026285842809, 539043082132918379, 570108181108560929, 610147978081735109 k=13 s=48 B={0 2 12 14 18 20 24 30 32 38 42 44 48} 10527733922579, 15991086371740199, 22443709342850669, 69759046409087909, 94415460183744419, 164873121596539229, 197053322268438509, 212971209388223159, 215768926871613989, 248170682800139819, 270109976153617319, 326374793491266239, 341896216415143109, 341987213500572359, 362035072661912369, 401062754451879239, 441180406661470349, 450928996714672349, 503035098004929209, 533306698691196149 k=14 s=50 B={0 2 6 8 12 18 20 26 30 32 36 42 48 50} 1 11 2 21817283854511261 (17 digits, 1982, D. Betsis & S. Säfholm) 3 841262446570150721 (18 digits, 2006, Vladimir Vlesycit) 4 1006587882969594041 (19 digits, 2006, Vladimir Vlesycit) 5 2682372491413700201 (2006, Vladimir Vlesycit) 6 5009128141636113611 (1996, TF, M500 153) 7 7126352574372296381 (2006, Vladimir Vlesycit) 8 7993822923596334941 (2006, Vladimir Vlesycit) 9 12870536149631655611 (20 digits, 2006, Vladimir Vlesycit) 10 15762479629138737611 (2006, Vladimir Vlesycit) 11 22811391659969177381 (2009, Joerg Waldvogel) 12 25593071201116914611 (2009, Joerg Waldvogel) 13 25908823282383706781 (2009, Joerg Waldvogel) 14 42408988436795325461 (2009, Joerg Waldvogel) 15 44360646117391789301 (2001, Tom Hadley) 16 54533311498517797061 (2009, Joerg Waldvogel) 17 54872625531142644341 (2009, Joerg Waldvogel) 18 55603189618237916771 (2009, Joerg Waldvogel) 19 60815831883653681951 (2009, Joerg Waldvogel) 20 65419797485410082831 (2009, Joerg Waldvogel) 21 66902362052699528471 (2009, Joerg Waldvogel) 22 67071371902365125411 (2009, Joerg Waldvogel) 23 68579452055759680151 (2009, Joerg Waldvogel) 24 70155086239067224511 (2009, Joerg Waldvogel) 25 72067980410977986551 (2009, Joerg Waldvogel) *26 80846604473064395081 (2009, Joerg Waldvogel) 27 85314045295767423251 (2009, Joerg Waldvogel) *28 85542881495337691541 (2009, Joerg Waldvogel) 29 88283061705894830891 (2009, Joerg Waldvogel) 30 93187668843868223741 (2009, Joerg Waldvogel) 31 94334865441620058371 (2009, Joerg Waldvogel) 32 98328854344531485791 (2009, Joerg Waldvogel) 33 98733575252292966101 (2009, Joerg Waldvogel) *34 113615698477681825541 (21 digits, 2009, Joerg Waldvogel) 35 116412111819886888121 (2009, Joerg Waldvogel) *36 116591588863353569081 (2009, Joerg Waldvogel) 37 121842488991384286241 (2009, Joerg Waldvogel) 38 122824498715815291511 (2009, Joerg Waldvogel) 39 125783435625318576221 (2009, Joerg Waldvogel) 40 129028836259200732971 (2009, Joerg Waldvogel) 41 137073444791483975891 (2009, Joerg Waldvogel) *42 140245881111654813611 (2009, Joerg Waldvogel) 43 145924412230165103651 (2009, Joerg Waldvogel) 44 148052012497754750501 (2009, Joerg Waldvogel) 45 156498140336020628261 (2009, Joerg Waldvogel) 46 160192637948100132491 (2009, Joerg Waldvogel) 47 163499038187940249911 (2009, Joerg Waldvogel) 48 163863898625903741381 (2009, Joerg Waldvogel) 49 164765327947624839551 (2009, Joerg Waldvogel) 50 166629298858950953141 (2009, Joerg Waldvogel) 51 169120923765003555461 (2009, Joerg Waldvogel) 52 178594761946767821771 (2009, Joerg Waldvogel) 53 189781953117081049841 (2009, Joerg Waldvogel) 54 193857647362637413271 (2009, Joerg Waldvogel) 55 198347648512106968991 (2009, Joerg Waldvogel) *56 204185491710368653601 (2009, Joerg Waldvogel) 57 209011112165465264771 (2009, Joerg Waldvogel) k=14 s=50 B={0 2 8 14 18 20 24 30 32 38 42 44 48 50} 1 79287805466244209 (17 digits, 1982, D. Betsis & S. Säfholm) 2 2714623996387988519 (19 digits, 2006, Vladimir Vlesycit) 3 5012524663381750349 (2006, Vladimir Vlesycit) 4 6120794469172998449 (1996, TF) 5 6195991854028811669 (2006, Vladimir Vlesycit) 6 6232932509314786109 (2006, Vladimir Vlesycit) 7 6808488664768715759 (1996, TF) 8 10756418345074847279 (20 digits,1997, TF) 9 11319107721272355839 (1997, TF) 10 12635619305675250719 (2006, Vladimir Vlesycit) 11 14028155447337025829 (2006, Vladimir Vlesycit) *12 14094050870111867489 (2006, Vladimir Vlesycit) 13 14603617704434643719 (2006, Vladimir Vlesycit) 14 14777669568340323479 (2006, Vladimir Vlesycit) 15 15420967329931107779 (2006, Vladimir Vlesycit) *16 16222575536498135639 (2006, Vladimir Vlesycit) 17 16624441191356313149 (2006, Vladimir Vlesycit) 18 17367037621075657349 (2006, Vladimir Vlesycit) 19 19289576760019250519 (2009, Joerg Waldvogel) 20 21392267423397385349 (2009, Joerg Waldvogel) 21 22322953993626430979 (2009, Joerg Waldvogel) 22 23259964787831666729 (2009, Joerg Waldvogel) 23 24257496589766344739 (2009, Joerg Waldvogel) 24 25465166216735269859 (2009, Joerg Waldvogel) 25 26531763621312864749 (2009, Joerg Waldvogel) *26 27367669288651556699 (2009, Joerg Waldvogel) 27 30617981105711113949 (2009, Joerg Waldvogel) 28 33557907113525709749 (2009, Joerg Waldvogel) 29 35863507147601267969 (2009, Joerg Waldvogel) 30 39684005203445427659 (2009, Joerg Waldvogel) 31 40850935638111603839 (2009, Joerg Waldvogel) 32 43075812519432827969 (2009, Joerg Waldvogel) 33 47498206250008368689 (2009, Joerg Waldvogel) 34 47583224994905364599 (2009, Joerg Waldvogel) *35 47972909762951107769 (2009, Joerg Waldvogel) *36 49198349218890756719 (2009, Joerg Waldvogel) 37 50079448898342988959 (2009, Joerg Waldvogel) 38 61544774858546043899 (2009, Joerg Waldvogel) 39 75358632265507015259 (2009, Joerg Waldvogel) 40 76506482782900995659 (2009, Joerg Waldvogel) 41 82812543147329880719 (2009, Joerg Waldvogel) *42 83522578855892290949 (2009, Joerg Waldvogel) 43 92304918286829362949 (2009, Joerg Waldvogel) 44 92414752514930310299 (2009, Joerg Waldvogel) 45 107329639491855415469 (21 digits, 2009, Joerg Waldvogel) 46 118830936551237785709 (2009, Joerg Waldvogel) 47 122338154285238166109 (2009, Joerg Waldvogel) 48 126601158086598870509 (2009, Joerg Waldvogel) 49 132038405083066376729 (2009, Joerg Waldvogel) 50 141372594058656295169 (2009, Joerg Waldvogel) *51 141851207652098108999 (2009, Joerg Waldvogel) 52 147512956871684579009 (2009, Joerg Waldvogel) 53 148987416994433071889 (2009, Joerg Waldvogel) 54 149205017745130902389 (2009, Joerg Waldvogel) 55 155607030758019625619 (2009, Joerg Waldvogel) 56 156337953807179268959 (2009, Joerg Waldvogel) 57 158405158462698882569 (2009, Joerg Waldvogel) 58 159106208338543350719 (2009, Joerg Waldvogel) 59 165106484117326676759 (2009, Joerg Waldvogel) *60 166270537650069307289 (2009, Joerg Waldvogel) 61 167988286274632568579 (2009, Joerg Waldvogel) 62 169736979674544356909 (2009, Joerg Waldvogel) 63 172547315507565683519 (2009, Joerg Waldvogel) 64 174429595099456671809 (2009, Joerg Waldvogel) 65 176587525606789261799 (2009, Joerg Waldvogel) 66 177038805029012571479 (2009, Joerg Waldvogel) 67 186153638297282831009 (2009, Joerg Waldvogel) 68 186389225345734315739 (2009, Joerg Waldvogel) 69 188421380809838711369 (2009, Joerg Waldvogel) 70 192812711703659071859 (2009, Joerg Waldvogel) *71 195690251815182929639 (2009, Joerg Waldvogel) 72 199738213891893771929 (2009, Joerg Waldvogel) 73 202646150547250214639 (2009, Joerg Waldvogel) 74 203862456461972513039 (2009, Joerg Waldvogel) 75 204821759330868384539 (2009, Joerg Waldvogel) k=15 s=56 B={0 2 6 8 12 18 20 26 30 32 36 42 48 50 56} 1 11 2 44360646117391789301 (20 digits, 2001, Tom Hadley) 3 80846604473064395081 (2009, Joerg Waldvogel) 4 85542881495337691541 (2009, Joerg Waldvogel) 5 113615698477681825541 (21 digits 2009, Joerg Waldvogel) 6 116591588863353569081 (2009, Joerg Waldvogel) 7 140245881111654813611 (2009, Joerg Waldvogel) 8 204185491710368653601 (2009, Joerg Waldvogel) k=15 s=56 B={0 2 6 12 14 20 24 26 30 36 42 44 50 54 56} 1 17 2 17905159760365247387 (20 digits, 2001, Jim Morton) 3 44333554877816671757 (2009, Joerg Waldvogel) 4 64777971223063936127 (2009, Joerg Waldvogel) 5 84528323459951417987 (2009, Joerg Waldvogel) 6 90798766993022298227 (2009, Joerg Waldvogel) 7 151477098804870766217 (21 digits, 2009, Joerg Waldvogel) 8 190685406656969508497 (2009, Joerg Waldvogel) 9 191219032841144805437 (2009, Joerg Waldvogel) k=15 s=56 B={0 2 6 12 14 20 26 30 32 36 42 44 50 54 56} 1 1158722981124148367 (19 digits, 2009, Joerg Waldvogel) 2 1240068005144831867 (1999, Joerg Waldvogel) 3 2243290806352501637 (2009, Joerg Waldvogel) 4 3172206835341609797 (2009, Joerg Waldvogel) 5 8402171449067476007 (2009, Joerg Waldvogel) &6 9422138120166964847 (2009, Joerg Waldvogel) 7 16485850001899818467 (20 digits, 2009, Joerg Waldvogel) 8 16679857156627718057 (2009, Joerg Waldvogel) 9 26082001869529405727 (2009, Joerg Waldvogel) 10 28025081332808782877 (2009, Joerg Waldvogel) &11 30304192466683200797 (2009, Joerg Waldvogel) 12 34827058725936837287 (2009, Joerg Waldvogel) 13 36351118555624575707 (2009, Joerg Waldvogel) 14 44715422969365380887 (2009, Joerg Waldvogel) 15 45912500276798107607 (2009, Joerg Waldvogel) 16 47710850533373130107 (1997, Tony Forbes) 17 47939384425506704057 (1999, Joerg Waldvogel) 18 48902741158702754327 (2009, Joerg Waldvogel) 19 50035247022695224217 (2009, Joerg Waldvogel) 20 63237840473489561807 (2009, Joerg Waldvogel) 21 65131261663903414907 (1999, Joerg Waldvogel) 22 71606347081895458487 (1999, Joerg Waldvogel) 23 87827558613079768487 (2009, Joerg Waldvogel) 24 105151328771084515847 (21 digits, 1999, Joerg Waldvogel) 25 107862607835977274207 (1997, TF) 26 112826662375907079527 (2009, Joerg Waldvogel) &27 122500782968891019347 (2009, Joerg Waldvogel) 28 126132544561513685927 (2009, Joerg Waldvogel) 29 135606914709645689237 (2009, Joerg Waldvogel) 30 138220583274981803327 (2009, Joerg Waldvogel) 31 191032326406786434497 (2009, Joerg Waldvogel) 32 205700275761622834847 (1997, TF) k=15 s=56 B={0 6 8 14 20 24 26 30 36 38 44 48 50 54 56} 1 14094050870111867483 (20 digits, 2007, Jens Kruse Andersen) 2 16222575536498135633 (2009, Joerg Waldvogel) 3 27367669288651556693 (2009, Joerg Waldvogel) 4 47972909762951107763 (2009, Joerg Waldvogel) 5 49198349218890756713 (2009, Joerg Waldvogel) 6 83522578855892290943 (2009, Joerg Waldvogel) 7 141851207652098108993 (21 digits, 2009, Joerg Waldvogel) 8 166270537650069307283 (2009, Joerg Waldvogel) 9 195690251815182929633 (2009, Joerg Waldvogel) k=16 s=60 B={0 4 6 10 16 18 24 28 30 34 40 46 48 54 58 60} 1 13 2 695874886175252911063 (21 digits, 1996, Joerg Waldvogel) 3 1567582627835236839763 (22 digits, 2009, Joerg Waldvogel) 4 1750052554011927712483 (2009, Joerg Waldvogel) 5 2257588388550898970503 (2009, Joerg Waldvogel) 6 3789227751026345304613 (2009, Joerg Waldvogel) 7 4654682384109074514133 (2009, Joerg Waldvogel) 8 5022156579757255625623 (2009, Joerg Waldvogel) 9 13599236099159166553033 (2009, Joerg Waldvogel) 10 29894522822363684652103 (23 digits, 2009, Joerg Waldvogel) 11 35718904544536715448883 (2009, Joerg Waldvogel) 12 42421183685552747462323 (2009, Joerg Waldvogel) 13 47624415490498763963983 (2001, Peter Leikauf and Joerg Waldvogel) 14 50069823850049036630533 (2009, Joerg Waldvogel) 15 56294926786180569503953 (2009, Joerg Waldvogel) 16 60877851518090858117803 (2009, Joerg Waldvogel) 17 66871135379953148611303 (2009, Joerg Waldvogel) 18 70743491366529526461853 (2009, Joerg Waldvogel) 19 72039555441202354852783 (2009, Joerg Waldvogel) 20 72939169778564978895943 (2009, Joerg Waldvogel) 21 78314167738064529047713 (2001, Peter Leikauf and Joerg Waldvogel) 22 79415821643818505392483 (2009, Joerg Waldvogel) 23 80755924819458605984203 (2009, Joerg Waldvogel) 24 81433995543774820773763 (2009, Joerg Waldvogel) 25 83405687980406998933663 (2001, Peter Leikauf and Joerg Waldvogel) 26 93091355499511979496823 (2009, Joerg Waldvogel) 27 94045986419037158522953 (2009, Joerg Waldvogel) 28 94975866735959660878363 (2009, Joerg Waldvogel) 29 97907646478336552814893 (2009, Joerg Waldvogel) 30 99517467925153764522973 (2009, Joerg Waldvogel) k=16 s=60 B={0 2 6 12 14 20 26 30 32 36 42 44 50 54 56 60} 1 47710850533373130107 (20 digits, 1997, TF and Joerg Waldvogel) 2 347709450746519734877 (21 digits, 1996, Joerg Waldvogel) 3 1099638576123052218257 (22 digits, 1996, Joerg Waldvogel) 4 1169914227530138703617 (1996, Joerg Waldvogel) 5 1522014304823128379267 (1997, Tony Forbes) 6 1620784518619319025977 (1997, Joerg Waldvogel) 7 2639154464612254121537 (1998, Joerg Waldvogel) 8 3259125690557440336637 (1997, Tony Forbes) 9 9042634271485192050677 (2009, Joerg Waldvogel) 10 9239395687646993061197 (2009, Joerg Waldvogel) 11 15571053758048293307807 (23 digits, 2009, Joerg Waldvogel) 12 20628149050698694668167 (2009, Joerg Waldvogel) 13 20947353617877810296177 (1999, Tony Forbes) 14 23182160505954925788317 (2009, Joerg Waldvogel) 15 27814116054901200587567 (2009, Joerg Waldvogel) 16 30406149669349341460577 (2009, Joerg Waldvogel) 17 31607383424682394081757 (2009, Joerg Waldvogel) 18 34254730511961158822627 (2009, Joerg Waldvogel) 19 40675973411840597813987 (2009, Joerg Waldvogel) 20 41459159264655740911307 (2009, Joerg Waldvogel) 21 49798002215047773229547 (2009, Joerg Waldvogel) 22 53284658140441622257367 (2009, Joerg Waldvogel) 23 61196813406933554172827 (2009, Joerg Waldvogel) 24 62564565965531138812307 (2009, Joerg Waldvogel) 25 67089635430601104554237 (2009, Joerg Waldvogel) 26 73377493322819357084417 (2009, Joerg Waldvogel) 27 75356498508757177334627 (2009, Joerg Waldvogel) 28 80444001154123052328887 (2009, Joerg Waldvogel) 29 81044576768686984666217 (2009, Joerg Waldvogel) 30 82069568461982909560757 (2009, Joerg Waldvogel) 31 86042715202634832665027 (2009, Joerg Waldvogel) 32 88077044957120569823477 (2009, Joerg Waldvogel) 33 89172493451857275102647 (2009, Joerg Waldvogel) 34 91576165734484982223677 (2009, Joerg Waldvogel) 35 93954295353554241598997 (2009, Joerg Waldvogel) 36 97692369541610844803807 (2009, Joerg Waldvogel) 37 98659593721870389301097 (2009, Joerg Waldvogel) k=17 s=66 B={0 4 10 12 16 22 24 30 36 40 42 46 52 54 60 64 66} 1 734975534793324512717947 (24 digits, 2009, Joerg Waldvogel) 2 753314125249587933791677 (2009, Joerg Waldvogel) k=17 s=66 B={0 4 6 10 16 18 24 28 30 34 40 46 48 54 58 60 66} 1 13 2 47624415490498763963983 (23 digits, 2001, Peter Leikauf and Joerg Waldvogel) 3 78314167738064529047713 (2001, Peter Leikauf and Joerg Waldvogel) 4 83405687980406998933663 (2001, Peter Leikauf and Joerg Waldvogel) 5 110885131130067570042703 (24 digits, 2001, Peter Leikauf and Joerg Waldvogel) 6 163027495131423420474913 (2001, Peter Leikauf and Joerg Waldvogel) k=17 s=66 B={0 6 8 12 18 20 26 32 36 38 42 48 50 56 60 62 66} 1 1620784518619319025971 (22 digits, 1997, Joerg Waldvogel) 2 2639154464612254121531 (1998, Joerg Waldvogel) 3 3259125690557440336631 (22 digits, 1998, Tony Forbes) 4 124211857692162527019731 (24 digits, 2001, Peter Leikauf and Joerg Waldvogel) k=17 s=66 B={0 2 6 12 14 20 24 26 30 36 42 44 50 54 56 62 66} 1 17 2 37630850994954402655487 (23 digits, 3 53947453971035573715707 (1998, Tony Forbes) 4 174856263959258260646207 (2001, Peter Leikauf and Joerg Waldvogel) 5 176964638100452596444067 (2001, Peter Leikauf and Joerg Waldvogel) 6 207068890313310815346497 (24 digits, Peter Leikauf and Joerg Waldvogel) 7 247620555224812786876877 (2001, Peter Leikauf and Joerg Waldvogel) 8 322237784423505559739147 (2001, Peter Leikauf and Joerg Waldvogel) k=18 s=70 B={0 4 10 12 16 22 24 30 36 40 42 46 52 54 60 64 66 70} 1 2845372542509911868266807 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (2000, Joerg Waldvogel & Peter Leikauf) k=18 s=70 B={0 4 6 10 16 18 24 28 30 34 40 46 48 54 58 60 66 70} 1 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 2 1906230835046648293290043 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66, 70 (2001, Joerg Waldvogel & Peter Leikauf) k=19 s=76 B={0 6 10 16 18 22 28 30 36 42 46 48 52 58 60 66 70 72 76} 1 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) 2 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) k=19 s=76 B={0 4 6 10 16 22 24 30 34 36 42 46 52 60 64 66 70 72 76} 1 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 k=19 s=76 B={0 4 6 10 12 16 24 30 34 40 42 46 52 54 60 66 70 72 76} No known examples of this pattern k=19 s=76 B={0 4 6 10 16 18 24 28 30 34 40 46 48 54 58 60 66 70 76} 1 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 k=20 s=80 B={0 2 8 12 14 18 24 30 32 38 42 44 50 54 60 68 72 74 78 80} 1 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 k=20 s=80 B={0 2 6 8 12 20 26 30 36 38 42 48 50 56 62 66 68 72 78 80} No known examples of this pattern Known prime k-tuplets for k = 21, 22, ..., 78: k=21 s=84 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 k=22 s=90 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 k=22 s=90 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 k=23 s=94 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 k=24 s=100 none k=25 s=110 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 k=26 s=114 none k=27 s=120 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 k=28 s=126 none k=29 s=130 none k=30 s=136 none k=31 s=140 none k=32 s=146 none k=33 s=152 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 k=34 s=156 none k=35 s=158 none k=36 s=162 none k=37 s=168 none k=38 s=176 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 k=39 s=182 none k=40 s=186 none k=41 s=188 none k=42 s=196 none k=43 s=200 none k=44 s=210 none k=45 s=212 none k=46 s=216 none k=47 s=226 none k=48 s=236 none k=49 s=240 none k=50 s=246 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 k=51 s=252 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 k=51 s=252 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 k=52 s=254 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 k=53 s=264 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 k=54, ..., 76 none k=77 s=420 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 k=77 s=420 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 k=77 s=420 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 k=78 s=422 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 Last updated: 9 June 2012