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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-Sep-09


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 946
+12 7367 949
+13 7387 951
+14 7401 953
+15 7429 956
+16 7441 958
+17 7445 959
+18 10031 1249
+19 10413 1293
+20 10423 1294
+21 10451 1298
+22 10543 1310
+23 10555 1311
+24 10821 1339
+25 10931 1351
+26 11045 1363 2006 *
+27 11575 1419  
+28 11577 1420
+29 11671 1429
+30 11773 1439
+31 13581 1637
+32 13609 1640
+33 14137 1696
+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 16629 1963
+42 16807 1981
+43 16857 1987
+44 16867 1988
+45 17165 2021
+46 17179 2023
+47 17275 2033
+48 18667 2179
+49 18709 2184
+50 18855 2197
+51 18889 2200
+52 18897 2201
+53 18967 2209
+54 18991 2212
+55 18999 2213
+56 19155 2228
+57 19201 2233
+58 19349 2249
+59 19357 2250
+60 19369 2651
+61 22917 2618
+62 22927 2620
+63 22991 2626
+64 23403 2671
+65 23409 2672
+66 23413 2673
+67 23465 2678
+68 23471 2679
+69 23507 2682
+70 23521 2684
+71 23529 2685
+72 24741 2809
+73 24837 2818
+74 24915 2826
k-p(w) width k year by
+75 25095 2844
+76 25291 2864
+77 25443 2881
+78 25513 2888
+79 25563 2893
+80 26549 2993
+81 26587 2997
+82 26623 3000
+83 26667 3005
+84 27387 3077
+85 27391 3078
+86 27519 3091
+87 27523 3092
+88 27609 3100
+89 27687 3107
+90 27727 3112
+91 28255 3168
+92 28259 3169
+93 28273 3170
+94 28377 3181
+95 28383 3182
+96 28511 3196
+97 29829 3327
+98 29905 3336
+99 29915 3337
+100 30001 3345
+101 30043 3349
+102 30069 3352
+103 30457 3391
+104 30487 3394
+105 30627 3407
+106 30827 3428
+107 31003 3447
+108 31829 3528
+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 35043 3849
+116 35255 3871
+117 35719 3912
+118 35725 3913
+119 35793 3920
+120 36425 3981
+121 36445 3983
+122 37825 4122
+123 37923 4132
+124 37939 4133
+125 38111 4148
+126 38147 4151
+127 38527 4186
+128 38535 4187
+129 38551 4189
+130 38591 4194
+131 39579 4294
+132 39599 4296
+133 39617 4298
+134 39815 4317
+135 40273 4363
+136 40299 4367
+137 40309 4368
+138 40319 4369
+139 40327 4370
+140 40333 4371
+141 40385 4376
+142 40403 4378
+143 40415 4379
+144 40421 4380
+145 41105 4445
+146 41439 4480
+147 41465 4483
+148 41503 4487
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
* -- created by Thomas J Engelsma from tuples submitted by Prof. Joerg Waldvogel
blank -- progress of searches by Thomas J Engelsma using sub-patterns from all existing tuples

The links to the raw data are to ASCII text files. The data is in half-difference format.

Graphics from the clip art site RetroKat
2008 Thomas J Engelsma