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The trophy case table below shows first known occurances of super-dense k-tuples.
The first column shows the number of additional primes that can be packed in the
interval, as a contradiction to a Hardy-Littlewood conjecture.

Additions to the table can be posted by sending the residue sets or a listing of prime positions
for the width to be added.  Please send any additions to tom(at)opertech.com

Last update : 12-Jun-17 (marked with "*")


Trophy Case

k-p(w) width k year by
+1 3159 447 see note
+2 3241 459 2004 TE
+3 3243 460
+4 4589 624
+5 4875 657
+6 4899 660
+7 5365 715  
+8 5371 716
+9 6541 853 2005 JW
+10 6647 866  

 

 

+11 * 7359 948
+12 7367 949
+13 7387 951
+14 7401 953
+15 7429 956
+16 7441 958
+17 7445 959
+18 * 8593 1087
+19 10413 1293
+20 10423 1294
+21 10451 1298
+22 10543 1310
+23 10555 1311
+24 10821 1339
+25 10931 1352
+26 * 11043 1363
+27 * 11053 1365
+28 11577 1420
+29 * 11665 1429
+30 * 11767 1439
+31 * 12369 1506
+32 13609 1640
+33 * 13675 1647
+34 14227 1706
+35 14229 1707
+36 14239 1708
+37 14301 1714
k-p(w) width k year by
+38 14317 1716  
+39 14373 1723
+40 14379 1724
+41 * 16593 1960
+42 * 16599 1961
+43 * 16629 1965
+44 * 16861 1988
+45 * 16867 1989
+46 17179 2023
+47 17275 2033
+48 * 17823 2091
+49 18709 2184
+50 * 18829 2196
+51 * 18835 2197
+52 * 18857 2199
+53 * 18883 2202
+54 * 18887 2203
+55 * 18897 2204
+56 * 18955 2211
+57 * 19067 2221
+58 * 19203 2234
+59 * 19349 2250
+60 * 19357 2251
+61 * 19367 2252
+62 22927 2620
+63 * 22957 2623
+64 23403 2671
+65 23409 2672
+66 23413 2673
+67 23465 2678
+68 23471 2679
+69 * 23493 2681
+70 23521 2684
+71 23529 2685
+72 24741 2809
+73 * 24835 2818
+74 * 24901 2825
k-p(w) width k year by
+75 * 25081 2843  
+76 * 25213 2857
+77 * 25407 2877
+78 * 25507 2888
+79 * 25519 2889
+80 * 25573 2994
+81 * 26531 2993
+82 * 26549 2995
+83 * 26615 3001
+84 * 26661 3006
+85 27391 3078
+86 27519 3091
+87 27523 3092
+88 27609 3100
+89 * 27683 3107
+90 * 27723 3112
+91 * 27731 3113
+92 * 28255 3169
+93 * 28269 3170
+94 * 28273 3171
+95 * 28381 3182
+96 28511 3196
+97 * 29785 3324
+98 * 29817 3327
+99 * 29829 3329
+100 * 29911 3338
+101 * 29971 3344
+102 * 30041 3350
+103 * 30057 3352
+104 * 30069 3354
+105 * 30613 3407
+106 * 30629 3408
+107 * 30827 3429
+108 * 31003 3448
+109 31837 3529
+110 31931 3537
+111 31945 3538
k-p(w) width k year by
+112 31951 3539  
+113 34111 3757
+114 34115 3758
+115 * 35005 3847
+116 * 35013 3848
+117 * 35021 3849
+118 * 35043 3852
+119 * 35727 3914
+120 * 36419 3981
+121 * 36425 3982
+122 * 36445 3984
+123 * 37825 4123
+124 * 37923 4133
+125 * 38095 4148
+126 * 38111 4149
+127 * 38147 4152
+128 * 38519 4187
+129 * 38531 4188
+130 * 38537 4189
+131 * 38555 4291
+132 * 39579 4295
+133 * 39601 4297
+134 * 39655 4302
+135 * 40215 4359
+136 * 40273 4364
+137 * 40299 4368
+138 * 40309 4369
+139 * 40319 4370
+140 * 40327 4371
+141 * 40333 4372
+142 * 40385 4377
+143 * 40403 4379
+144 * 40415 4380
+145 * 40421 4381
+146 * 41105 4446
+147 * 41439 4481
+148 * 41465 4484
The 447-tuple was independently found by Ralph Gasser/Prof. Joerg Waldvogel, Mischa Kenn, and Thomas J Engelsma
TE -- Thomas J Engelsma
JW-- Prof. Joerg Waldvogel
blank -- progress of searches by Thomas J Engelsma using sub-patterns from all existing tuples

Graphics from the clip art site RetroKat
2008,2017 Thomas J Engelsma