| Patterns_in_Primes Examples of mostly digit patterns collected by Harvey Heinz. |
| Primality Introductory text on the theory of prime numbers and number fields. Contains proofs of some important theorems including the fundamental theorem of arithmetic. [PDF] |
| Primality_Testing_with_Fermat\'s_Little_Theorem Test numbers for primality and pseudoprimality in Java. |
| Prime_k-tuplets Tony Forbes' extensive collection of special types of prime clusters. |
| The_Prime_Machine Explore interactively the Goldbach conjecture, the distribution of prime twins, the prime number theorem. |
| Prime_Music A musical piece translating prime factorizations to frequencies. Requires RealPlayer. |
| Prime_Numbers Short lesson. |
| Prime_Numbers_and_Factoring Links and references related to primes and factoring. |
| Prime_Numbers_List Browse all prime numbers of less than 10 digits. A prime number checker facility is also included. |
| The_Prime_Page Methods and definitions of finding prime numbers. |
| Radiant_Primes Visualization of prime numbers resembling an astronomical radiant or celestial pathway. |
| Some_Prime_Numbers Sieves and factoring for small numbers. Download primes below 8 billion. |
| Status_of_Search_for_Multifactorial_Primes An organized search for primes of the form (n)(n-k)(n-2k)...+/-1. Includes table of numbers searched and primes found. |
| Thomas_R__Nicely Includes twin prime counts, tables of first occurrence prime gaps, papers. |
| The_Time_Traveler A group with new propositions in various areas of number theory. |
| Visualizing_the_Distribution_of_Prime_Numbers Investigation into patterns in the distribution of the primes by visualizing them. |
| Wikipedia__Prime_Numbers Index to many prime-related articles. |
| World_of_Numbers__World_of_Palindromic_Primes Records, statistics, curios and puzzles about primes reading the same backwards. Compiled by Patrick De Geest. |
| Bipeds_and_Prime_Numbers Graduation address by former professor of philosophy Garrett Barden. Mentions primes as sign of human curiosity. (September 18, 2001) |
| Chair_of_General_Chemistry,_Altai_State_Technical_University Describes scientific work and educational programs, lists group publications. |
| Department_of_Chemistry Contains online course material and lists faculty members. |
| Earth_Measurement_Corp_ Near-surface geophysical data acquisition, processing and interpretation. |
| Electro-Seise,_Inc_ 3D airborne micro gravity/E surveys for subsurface hydrocarbons. |
| Fairfield_Industries_Incorporated Fairfield is a geophysical company known for its development and use of advanced technology for acquisition and processing, and its licensing of seismic data. |
| G_A__Ryzhikov_and_M_S__Biryulina,_Consultants Special processing for traditional marine data using "Partial Synergetic Imaging-", or PSI-criterion. |
| Getech Geophysical contractor providing gravity and magnetic services and products to the international oil and mining industry. Offices in Leeds and Houston. |
| Gibson_Consulting Interpretation of gravity & magnetics for hydrocarbon exploration. |
| Input_Output,_Inc_ Provider of land, marine and reservoir seismic instrumentation. Includes comprehensive data sheets. |
| Interactive_Interpretation_&_Training,_Inc_ Outsourcing geoscience for the upstream petroleum industry: geophysics, geology, petroleum engineering, geochemistry, workstation rental. Landmark and GeoQuest software. |
| JGI,_Inc_ Acquisition of seismic and other geophysical data, geothermal surveys and remote sensing. Japan based. |
| Meridian_Surveys Survey and geophysical consultants to the oil and gas industry. |
| Montason_Exploration_Inc_ Geophysical acquisition, processing and interpretation. Electromagnetic technology produces virtual subsurface conductivity log. |
| Oilsearch_Plc_-_Airborne_Geophysical_Oil_Survey_Company Airborne geophysical oil survey company specialising in remote fluorescence sensing and magnetic data recovery. |
| PGS_Geophysical Onshore and marine seismic acquisition and analysis. Reservoir imaging, characterization and monitoring. Multiclient 3D data library. |
| Reid_Geophysics Offering gravity and magnetic survey specification, processing and interpretation. Proprietary software. |
| RGR__Method_of_Effective_Specular_Points A novel approach to Reflector Geometry Reconstruction: proper assimilation of prestacked multioffset traveltimes, macro-model-independent seismic reflection (depth) imaging. |
| Seismic_Consultants_Group_Pty__Ltd_ Seismic Consultants Group (SCG) provides personnel for field supervision and quality control of geophysical surveys to the oil and gas exploration industry worldwide |
| Seismic_Imaging_Research Research reports on subsalt imaging, depth migration, velocity model building, seismic modeling, converted mode energy and overpressure prediction. |
| SeismicDocs Online professional geophysical consulting service. |
| Seismograph_Rental_,_seismic_tomography_services Passive seismic tomography services for oil exploration. Seismograph and vibroseis rental. Pipeline pollution early warning systems. Seismological investigations. Crosshole seismic tomography. |
( 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 : 9-Nov-08
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 identified all k-tuple patterns in intervals of
3 to 2203.
Using the exhaustive search results the lower bound can be extended to 2227.
2227 <= y <= 3159
lower bound of 2227 <= y
(Nov 08)
upper bound of y <= 3159 (Feb 05)
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
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 325.
These k-values correspond to widths of 3 through 2203.
For k-values of 326 and larger, one or more variations of the listed k-tuple exist.
K-tuple patterns found
Current number of variations : 1,727,880 (widths of 1 to 5000)
Update History
Patterns from exhaustive search
thru width of 2203
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
k
w
k-p(w)
vars.
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
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
k
w
k-p(w)
vars.
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
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
k
w
k-p(w)
vars.
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
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
