ktuple1.jpg (9192 bytes)
( A gift from my wife )

K-TUPLE

Permissible Patterns

"Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have no reason to believe that it is a mystery into which the mind will ever penetrate."  Leonhard Euler

"A mathematician is a machine for turning coffee into theorems."  Paul Erdös

findings using an exhaustive search
last update : 12-Sep-09

by : Thomas J Engelsma
Send comments, suggestions, or requests to
tom(at)opertech.com

Special thanks to Prof. Joerg Waldvogel and Mischa Kenn

"A mathematician is a blind man in a dark room looking for a black cat which isn't there."  Charles Darwin

Tables and information concerning the Hardy-Littlewood conjecture that
p(x+y) -  p(x) <= p (y)

This conjecture fails if the k-tuple conjecture is true with a value of y = 3159.
An admissible k-tuple of 447 primes can be created in an interval of 3159 integers,
while p(3159) = 446.

Exhaustive searching has verified the Hardy-Littlewood conjecture is true for intervals up to 2529.
Exhaustive searching has identified all maximum density k-tuple patterns in intervals of 3 to 2331.

2529 < y <= 3159

lower bound of  2529 < y (Apr 09)
upper bound of  y <= 3159 (Feb 05)

Verified as maximum density
p(x+1417) - p(x) = p(1417) = 223
p(x+2529) - p(x) = p(2529) = 369
p(x+2655) - p(x) = p(2655) = 384

Patterns exist - not verified as maximum
p(x+3153) - p(x) = p(3153) = 446
p(x+3157) - p(x) = p(3157) = 446

List of residues for the 447-tuples of width 3159.
Calculations on the first occurrence of 447-tuples of width 3159.

Plot of k-p(w) versus width
Plot of k/p(w/2) versus ln(w)
Trophy Case - Table of first known widths for super-dense constellations
Linked Summary of k-tuple information up to width of 42741
Also, Summary as an entire text document  (734k)

Papers on the
K-Tuples
Conjecture

-
-
-
 Permissible Patterns of Primes (Sep 2009)
 Primes in a Fixed Interval 
 Entropy of the Primes

Other K-tuple information:

Dense Admissible Sets by  Daniel M. Gordon and Gene Rodemich
Admissible Prime Constellations by  Tomas Oliveira e Silva
Primes in Tuples I by D.A. Goldston, J. Pintz, and Y. Yildirim
Prime Constellation Records by Jens Kruse Anderen
Definition of  k-tuple entry from Prime Pages.
Definition of  k-tuple entry from World of Mathematics.

Sequence ID A020497 from On-Line Encyclopedia of Integer Sequences.

Problem #47 on the Prime Puzzle and Problems page.
List of residues for 50-tuples, 100-tuples, 150-tuples, and 200-tuples.

Using an exhaustive search program, written in assembler, ALL minimum
width k-tuple variations have been identified for k-values of 1 through 342.
These k-values correspond to widths of 3 through 2331.
For k-values of 343 and larger, one or more variations of the listed k-tuple exist.

K-tuple patterns found

Current number of variations : 1,722,826 (widths of 1 to 5000)
Update History
Patterns from exhaustive search
thru width of 2331
Patterns known to exist
'a' initial pattern by Ralph Gasser/Prof. Joerg Waldvogel
'b' initial patterns by Mischa Kenn
k w k-p(w) vars.
2 3 0 1
3 7 -1 2
4 9 0 1
5 13 -1 2
6 17 -1 1
7 21 -1 2
8 27 -1 3
9 31 -2 4
10 33 -1 2
11 37 -1 2
12 43 -2 2
13 49 -2 6
14 51 -1 2
15 57 -1 4
16 61 -2 2
17 67 -2 4
18 71 -2 2
19 77 -2 4
20 81 -2 2
21 85 -2 2
22 91 -2 4
23 95 -1 2
24 101 -2 4
25 111 -4 18
26 115 -4 2
27 121 -3 8
28 127 -3 10
29 131 -3 2
30 137 -3 2
31 141 -3 2
32 147 -2 4
33 153 -3 14
34 157 -3 20
35 159 -2 2
36 163 -2 2
37 169 -2 2
38 177 -2 6
39 183 -3 26
40 187 -2 26
41 189 -1 8
42 197 -3 2
43 201 -3 6
44 211 -3 18
45 213 -2 4
46 217 -1 4
47 227 -2 4
48 237 -3 2
49 241 -4 2
50 247 -3 22
51 253 -3 22
52 255 -2 2
53 265 -3 2
54 271 -4 26
55 273 -3 6
56 279 -3 6
57 283 -4 2
58 289 -3 2
59 301 -3 4
60 305 -2 2
61 311 -3 2
62 321 -4 6
63 325 -3 2
64 331 -3 2
65 337 -3 2
66 343 -2 2
67 351 -3 18
68 357 -3 2
69 367 -4 20
70 371 -3 2
71 379 -4 2
72 385 -4 2
73 391 -4 10
74 393 -3 2
75 399 -3 14
76 411 -4 14
77 421 -5 40
78 423 -4 8
79 427 -3 2
80 433 -4 14
81 439 -4 14
82 447 -4 16
83 451 -4 4
84 453 -3 2
85 463 -5 2
86 471 -5 60
87 477 -4 50
88 483 -4 2
89 487 -4 2
90 495 -4 2
91 505 -5 16
92 507 -4 2
93 513 -4 18
94 517 -3 12
95 519 -2 4
96 531 -3 4
97 537 -2 4
98 547 -3 4
99 553 -2 4
100 559 -2 4
101 573 -4 32
102 577 -4 8
103 579 -3 2
104 591 -3 46
105 601 -5 486
106 603 -4 50
107 607 -4 18
108 613 -4 24
109 617 -4 2
110 629 -4 6
111 635 -4 2
112 641 -4 4
113 647 -5 4
114 655 -5 10
115 657 -4 2
116 663 -5 4
117 673 -5 2
118 681 -5 4
119 687 -5 2
120 693 -5 2
121 703 -5 2
122 709 -5 4
123 715 -4 2
124 723 -4 2
125 733 -5 16
126 741 -5 26
127 747 -5 28
128 751 -5 6
129 761 -6 6
130 769 -6 2
131 775 -6 16
132 781 -5 30
133 785 -4 6
134 795 -4 2
135 805 -4 22
136 809 -4 4
137 813 -4 20
138 817 -3 18
139 819 -2 8
140 829 -5 6
141 841 -5 12
142 843 -4 6
143 849 -3 6
144 857 -4 24
145 865 -5 12
146 873 -4 2
147 879 -4 34
148 883 -5 18
149 893 -5 18
150 903 -4 22
151 909 -4 12
152 913 -4 10
153 927 -4 4
154 931 -4 8
155 935 -3 4
156 947 -5 10
157 953 -5 2
158 961 -4 2
159 971 -5 22
160 975 -4 20
161 987 -5 38
162 991 -5 14
163 999 -5 6
164 1003 -4 6
165 1013 -5 2
166 1023 -6 20
167 1027 -5 2
168 1033 -6 4
169 1037 -5 2
170 1045 -5 2
171 1051 -6 2
172 1059 -5 6
k w k-p(w) vars.
173 1067 -6 6
174 1071 -6 6
175 1075 -5 6
176 1083 -4 14
177 1087 -4 14
178 1105 -7 6
179 1111 -7 6
180 1121 -7 4
181 1125 -7 6
182 1131 -7 6
183 1143 -6 8
184 1147 -5 4
185 1151 -5 2
186 1163 -6 2
187 1169 -5 2
188 1177 -5 24
189 1183 -5 6
190 1189 -5 4
191 1195 -5 8
192 1201 -5 12
193 1205 -4 4
194 1211 -3 4
195 1219 -4 4
196 1231 -6 26
197 1239 -6 408
198 1259 -7 24
199 1263 -6 16
200 1267 -5 8
201 1275 -4 14
202 1281 -5 8
203 1291 -7 8
204 1303 -9 30
205 1309 -9 2
206 1317 -8 108
207 1321 -9 88
208 1329 -9 4
209 1333 -8 2
210 1339 -7 4
211 1345 -6 6
212 1351 -5 10
213 1353 -4 2
214 1359 -3 2
215 1365 -3 4
216 1371 -3 4
217 1375 -3 2
218 1381 -3 2
219 1387 -2 2
220 1393 -1 2
221 1405 -1 8
222 1413 -1 10
223 1417 0 8
224 1433 -3 8
225 1441 -3 4
226 1449 -3 4
227 1457 -4 20
228 1463 -4 16
229 1471 -4 12
230 1477 -3 2
231 1483 -4 8
232 1487 -4 2
233 1495 -5 2
234 1509 -5 4
235 1513 -5 2
236 1523 -5 2
237 1531 -5 2
238 1537 -4 4
239 1553 -6 14
240 1561 -6 114
241 1565 -5 20
242 1571 -6 12
243 1581 -6 2
244 1591 -6 16
245 1597 -6 6
246 1605 -6 12
247 1611 -7 8
248 1621 -9 6
249 1631 -9 12
250 1637 -9 10
251 1645 -8 10
252 1651 -7 6
253 1657 -7 24
254 1667 -8 72
255 1673 -8 46
256 1681 -7 152
257 1687 -6 92
258 1693 -6 46
259 1701 -7 92
260 1707 -6 46
261 1717 -6 88
262 1721 -6 6
263 1729 -6 20
264 1737 -6 10
265 1747 -7 12
266 1753 -7 50
267 1761 -7 38
268 1765 -6 12
269 1773 -5 68
270 1783 -6 96
271 1791 -7 50
272 1797 -6 4
273 1803 -6 4
274 1813 -6 44
275 1823 -6 28
276 1827 -5 4
277 1837 -5 32
278 1843 -4 56
279 1849 -4 24
280 1855 -3 8
281 1863 -3 8
282 1871 -4 12
283 1877 -5 12
284 1883 -5 12
285 1891 -5 20
286 1895 -4 8
287 1901 -4 8
288 1915 -5 8
289 1921 -4 8
290 1927 -3 8
291 1933 -4 16
292 1941 -3 16
293 1945 -2 16
294 1963 -3 8
295 1967 -2 2
296 1981 -3 58
297 1987 -3 34
298 1993 -3 98
299 2001 -4 80
300 2011 -5 258
301 2017 -5 24
302 2023 -4 158
303 2027 -4 6
304 2035 -4 6
305 2047 -4 20
306 2051 -3 4
307 2061 -3 4
308 2065 -3 4
309 2073 -3 6
310 2077 -2 6
311 2087 -4 4
312 2101 -5 36
313 2103 -4 2
314 2109 -3 2
315 2125 -4 2
316 2133 -5 1892
317 a 2137 -5 8
318 2145 -6 1182
319 2149 -5 724
320 2155 -5 632
321 2167 -5 174
322 2175 -4 2
323 2179 -4 2
324 2191 -3 260
325 2201 -2 170
326 2205 -2 66
327 2211 -2 66
328 a 2221 -3 6
329 2227 -2 292
330 2231 -1 146
331 2245 -3 770
332 2253 -3 506
333 2257 -2 48
334 2263 -1 1326
335 2267 -1 358
336 2271 -1 150
337 2287 -3 168
338 2299 -4 7940
339 2301 -3 40
340 2311 -4 40
341 2323 -3 2932
342 2329 -2 16
k w k-p(w) vars.
343 2341 -4 98
344 2343 -3 40
345 2355 -4 142
346 2359 -4 142
347 2365 -3 38
348 2377 -4 54
349 2383 -5 16
350 2389 -5 10
351 a 2395 -5 2
352 2401 -5 20
353 2407 -4 24
354 2411 -4 8
355 2419 -4 8
356 2441 -6 92
357 2445 -5 2
358 2453 -5 44
359 2461 -5 44
360 2475 -6 18
361 2481 -6 36
362 2485 -5 12
363 2491 -4 18
364 2497 -3 12
365 2503 -3 18
366 2517 -2 24
367 2521 -2 12
368 2523 -1 6
369 a 2529 0 10
370 2545 -2 12
371 2551 -3 28
372 2557 -3 16
373 a 2583 -3 18
374 2587 -2 18
375 2593 -3 12
376 2601 -2 76
377 2611 -2 156
378 2613 -1 52
379 2619 -1 40
380 2631 -1 38
381 2643 -1 38
382 2653 -1 74
383 a 2655 0 18
384 2661 -1 6
385 2675 -2 396
386 2685 -3 138
387 2691 -4 120
388 2703 -5 96
389 2707 -5 70
390 2713 -6 26
391 2719 -6 6
392 2731 -7 84
393 2733 -6 4
394 a 2737 -5 4
395 2751 -6 6
396 a 2757 -6 2
397 2763 -5 8
398 2773 -5 8
399 2787 -5 14
400 2791 -6 14
401 2797 -6 6
402 2807 -7 2
403 2819 -7 2
404 2823 -6 4
405 b 2827 -5 2
406 2841 -6 6
407 2849 -6 2
408 2855 -6 2
409 2861 -7 90
410 2865 -6 4
411 b 2869 -5 2
412 2887 -6 40
413 2891 -5 20
414 b 2901 -5 2
415 b 2907 -5 2
416 2917 -6 212
417 2925 -5 2
418 a 2929 -5 2
419 2939 -5 2
420 2951 -4 24
421 b 2957 -5 22
422 2967 -5 74
423 2973 -6 56
424 2977 -5 32
425 a 2991 -4 44
426 2995 -3 2
427 3001 -4 2
428 3017 -4 58
429 3025 -5 50
430 a 3031 -4 80
431 b 3035 -3 32
432 b 3039 -3 24
433 b 3043 -3 24
434 b 3053 -3 24
435 3073 -4 6318
436 3081 -4 88
437 3087 -4 88
438 3093 -4 82
439 3099 -3 70
440 3103 -2 70
441 3111 -2 314
442 3123 -3 48
443 3127 -2 48
444 3129 -1 22
445 3139 -1 22
446 3153 0 320
447 3159 1 12
448 3169 -1 20
449 3171 0 10
450 3177 1 10
451 3187 0 10
452 3193 0 10
453 3201 1 66
454 3211 0 132
455 3213 1 66
456 3219 1 66
457 3223 1 58
458 3237 1 116
459 3241 2 116
460 3243 3 58
461 3277 -1 58
462 3289 0 58
463 3303 -1 130
464 3307 -1 102
465 3309 0 40
466 3319 -1 82
467 3321 0 34
468 3327 0 24
469 3337 -1 24
470 3349 -2 24
471 3363 -3 24
472 3369 -2 2
473 3377 -3 10
474 3383 -2 10
475 3397 -3 256
476 3403 -2 228
477 3409 -2 200
478 3421 -2 288
479 3423 -1 126
480 3429 0 126
481 3433 0 126
482 3445 1 1024
483 3451 1 1024
484 3463 -1 1536
485 3469 -2 442
486 3481 -1 460
487 3489 0 14
488 3493 0 2
489 3501 0 12
490 3507 1 12
491 3515 1 2
492 3531 -1 2784
493 3535 -1 2046
494 3547 -3 424
495 3557 -3 424
496 3565 -3 216
497 3571 -3 2046
498 3583 -4 388
499 3587 -3 24
500 3595 -3 96
501 3605 -2 40
502 3613 -3 40
503 3623 -4 22
504 3631 -4 16
505 3641 -4 16
506 3649 -4 16
507 3655 -3 2
k w k-p(w) vars.
508 3661 -3 10
509 3667 -2 4
510 3671 -2 2
511 3685 -3 100
512 3691 -3 358
513 3697 -3 542
514 3701 -3 90
515 3707 -2 90
516 3723 -3 42
517 3727 -3 22
518 3739 -4 224
519 3741 -3 40
520 3751 -2 40
521 3763 -2 544
522 3771 -3 40
523 3781 -3 528
524 3787 -2 644
525 3789 -1 40
526 3811 -3 160
527 3813 -2 24
528 3817 -1 18
529 3837 -3 26
530 3841 -2 4
531 3843 -1 2
532 3865 -4 90
533 3873 -3 2
534 3879 -3 28
535 3883 -3 2
536 3897 -3 26
537 3901 -2 26
538 3911 -3 26
539 3919 -4 16
540 3921 -3 2
541 3927 -3 2
542 3939 -4 14
543 3949 -5 16
544 3951 -4 2
545 3957 -3 2
546 3961 -2 2
547 3969 -2 208
548 3979 -1 208
549 3997 -1 6
550 4003 -2 26
551 4015 -3 68
552 4021 -4 40
553 4025 -3 20
554 4037 -3 4
555 4043 -2 4
556 4051 -3 94
557 4053 -2 44
558 4059 -2 4
559 4063 -1 4
560 4083 -2 480
561 4089 -1 480
562 4093 -2 480
563 4099 -2 234
564 4111 -2 422
565 a 4113 -1 210
566 4117 0 2
567 4137 -2 2
568 4153 -3 1098
569 4161 -4 1260
570 4167 -3 1114
571 4177 -3 1110
572 4183 -2 20078
573 4189 -1 3136
574 4191 0 1504
575 a 4197 1 1504
576 4209 1 1502
577 4221 -1 1372
578 4237 -2 6576
579 4239 -1 2680
580 4249 -2 2724
581 4257 -2 2760
582 4261 -3 2750
583 4263 -2 36
584 4281 -3 32
585 4287 -3 32
586 4293 -3 8
587 4303 -3 40
588 4311 -2 2
589 4317 -1 2
590 4323 0 2
591 4327 0 2
592 4333 1 2
593 4341 0 4
594 4347 1 4
595 4351 1 4
596 4353 2 2
597 4363 1 2
598 4369 2 2
599 4377 2 34
600 4381 3 32
601 4399 2 32
602 4417 2 2
603 4423 1 32
604 4429 2 32
605 4441 2 34
606 4451 1 34
607 4457 1 2
608 4465 1 66
609 4469 2 32
610 4483 1 32
611 4493 1 34
612 4501 2 64
613 4505 3 30
614 4519 0 346
615 4527 0 316
616 4533 1 284
617 4547 1 2
618 4553 1 2
619 4557 2 2
620 4561 2 2
621 4571 2 2
622 4577 3 2
623 4585 3 62
624 4589 4 30
625 4603 2 30
626 4627 2 42
627 4639 1 30
628 4645 1 28
629 4651 0 28
630 4661 0 28
631 4669 0 218
632 4681 -1 426
633 4687 0 92
634 4693 0 4
635 4697 1 2
636 4705 1 2
637 4711 2 2
638 4723 1 2
639 4735 0 1048
640 4743 1 4
641 4747 2 4
642 4763 1 2010
643 4767 2 4
644 4775 3 1798
645 4789 1 1580
646 4799 0 400
647 4805 0 400
648 4813 0 376
649 4821 0 2
650 4831 0 28
651 4837 1 2
652 4843 2 24
653 4849 3 24
654 4857 4 2
655 4861 4 2
656 4871 4 2
657 4875 5 2
658 4885 5 28
659 4891 5 2
660 4899 6 24
661 4915 5 24
662 4925 5 6722
663 4933 4 2
664 4941 4 4
665 4945 4 2
666 4953 4 2
667 4957 4 2
668 4967 4 682
669 4973 3 2
670 4981 4 22
671 4991 4 2
672 4999 3 24

© 2008 Thomas J Engelsma