mersenneforum.org

mersenneforum.org (https://www.mersenneforum.org/index.php)
-   Prime Gap Searches (https://www.mersenneforum.org/forumdisplay.php?f=131)
-   -   GAPS BETWEEN PRIME PAIRS (Twin Primes) (https://www.mersenneforum.org/showthread.php?t=24303)

rudy235 2019-04-16 18:20

GAPS BETWEEN PRIME PAIRS (Twin Primes)
 
As we all know the twin primes are
{3,5} {5,7} {11.13} {17,19} {29,31} {41,43} {59,61} {71,73} {101,103} {107,109} {137,139} … [OEIS]A077800[/OEIS]

Chris Caldwell has a link to the first 10k Twin primes [URL="https://primes.utm.edu/lists/small/100ktwins.txt"]first element of twin primes[/URL]
Except for the first pair all the primes p, p+2 are the form 6k+1 and 6k-1

So we can adopt the convention of denoting a twin pair of primes by simple using the number k

Thus k=58 represents the twin primes 347, 349 or 6*58-1 and 6*58+1

Then we can create a sequence of all k's and have a good shorthand for all the pairs of twin primes (except for the aforementioned pair {3,5}

This sequence is [OEIS]A002822[/OEIS] 1,2,3,5,7,10,12,17,18,23,25,30,32,33,38,40,45,47,
52,58,70,72,77,87,95,100,103,107,110,135,137,138,
143,147,170,172,175,177,182,192,205,213,215,217,
220,238,242,247,248,268,270,278,283,287,298,312,
313,322,325

With the exception of the first, all of the members of this sequence are congruent to (0, 2 or 3 mod 5

So in a comparison to the "gap between primes" we now can establish gaps between contiguous pairs of primes. (of the form 6k [SUP]+[/SUP]/-1

The first gap of 1 appears at the start as we can see here

[code]
[B][SIZE="3"]k Gap[/SIZE][/B]
1 1
2 1
3 2
5 2
7 3
10 2
12 5
17 1
18 5
23 2
25 5
30 2
32 1
33 5
38 2
40 5
45 2
47 5
52 6
58 12
70 2
72 5
77 10
87 8
95 5
100 3
103 4
107 3
110 25
135 2
137 1
138 5
143 4
147 23
170 2
172 3
175 2
177 5
182 10
192 13
205 8
213 2
215 2
217 3
220 18
238 4
242 5
247 1
248 20
268 2
270 8
278 5
283 4
287 11
298 14
312 1
313 9
322 3
325 8
333 5
338[/code]

rudy235 2019-04-16 19:01

We can see a few things on this list of gaps.

First of all that all numbers seem to be represented ( we have at least 1 to 6 then 8 to 14 and 18, 20,23,25 )

What does a gap of 1 mean and are they infinite of those?

A gap of 1 between two pairs of twin primes represents a prime quadruplet For instance the gap of 1 after element 247 represents 6*247-1 and 6*247+1 which is a twin prime pair {1481,1483} and with the next closest pair of primes {1487, 1489} make a quadruplet.

As quadruplet primes [SPOILER]are theorized to be infinite[/SPOILER] the gaps of 1 would also be infinite.

Are all gaps represented? I believe so but, of course, this is an open question. I do not see any reason why a gap of 15 or of 7 might not exist and if someone with time and resourses makes a run up to twin primes under 100,000 I am confident that they should appear a few times. (I have only searched primes ≤ 2100 which is a paltry seach)

[code]k Gap
1 [B]1[/B] (first time)
2 1
3 [B] 2[/B] (first time)
5 2
7 [B]3[/B] (first time)
10 2
12 [B]5[/B] (first time)
17 1
18 5
23 2
25 5
30 2
32 1
33 5
38 2
40 5
45 2
47 5
52 [B] 6[/B] (first time)
58 [B]12[/B] (first time)
70 2
72 5
77 [B]10[/B] (first time)
87 [B] 8[/B] (first time)
95 5
100 3
103 [B]4[/B] (first time)
107 3
110 [B] 25[/B] (first time)
135 2
137 1
138 5
143 4
147 [B]23[/B] (first time)
170 2
172 3
175 2
177 5
182 10
192 [B]13[/B] (first time)
205 8
213 2
215 2
217 3
220 [B]18[/B] (first time)
238 4
242 5
247 1
248 [B]20[/B] (first time)
268 2
270 8
278 5
283 4
287 [B]11[/B] (first time)
298 [B]14[/B] (first time)
312 1
313 [B]9[/B] (first time)
322 3
325 8
333 5
338[/code]

ATH 2019-04-16 22:29

First occurrence gaps up to gap=1023.


[CODE]
gap k (before gap)
1 1
2 3
3 7
4 103
5 12
6 52
7 378
8 87
9 313
10 77
11 287
12 58
13 192
14 298
15 597
16 357
17 1075
18 220
19 3563
20 248
21 2042
22 800
23 147
24 3843
25 110
26 3257
27 2063
28 397
29 6458
30 1755
31 6227
32 1438
33 1507
34 5638
35 980
36 13372
37 2560
38 7637
39 6018
40 2438
41 6332
42 6088
43 14542
44 11833
45 2478
46 6692
47 2233
48 9105
49 6808
50 8432
51 23277
52 7968
53 6585
54 23133
55 13815
56 25347
57 13953
58 7462
59 35798
60 31972
61 25587
62 3090
63 17475
64 90423
65 9002
66 72942
67 19033
68 17850
69 32378
70 4377
71 12952
72 23693
73 48785
74 37058
75 14845
76 58077
77 35368
78 77697
79 18308
80 55143
81 33397
82 74400
83 4070
84 24168
85 20478
86 138187
87 80868
88 22890
89 56523
90 55632
91 37942
92 81448
93 44660
94 213103
95 97545
96 354662
97 27620
98 27977
99 383148
100 44905
101 125472
102 20013
103 76967
104 293123
105 10383
106 241847
107 91303
108 89567
109 91103
110 140420
111 129022
112 149863
113 130757
114 239678
115 126070
116 334862
117 100658
118 429182
119 304243
120 140795
121 614567
122 322018
123 199937
124 535353
125 73238
126 148897
127 135550
128 410312
129 217368
130 37355
131 264492
132 150433
133 187047
134 376973
135 500720
136 828307
137 156343
138 311532
139 767163
140 505115
141 569097
142 240345
143 41995
144 989058
145 116403
146 407412
147 308028
148 802070
149 989443
150 282408
151 707262
152 277258
153 234965
154 31318
155 114587
156 607952
157 292908
158 839762
159 410923
160 154445
161 518717
162 793900
163 526890
164 443443
165 437057
166 705787
167 632180
168 114742
169 1571558
170 1074442
171 2040992
172 799050
173 1042935
174 2320048
175 613727
176 322557
177 252443
178 1414700
179 1773728
180 302698
181 2253632
182 1038265
183 735717
184 1772778
185 608167
186 729092
187 364473
188 416322
189 3004313
190 2048442
191 880507
192 1052370
193 485555
194 1918663
195 857552
196 1402747
197 682880
198 1188775
199 2004258
200 340200
201 1511522
202 1678078
203 1335645
204 4055193
205 1722322
206 1372987
207 3142030
208 546497
209 543638
210 1330010
211 2007457
212 1338953
213 2109557
214 1627603
215 439318
216 4449232
217 901498
218 1416445
219 1886258
220 1793428
221 3411077
222 1580488
223 1720645
224 4460943
225 1982738
226 1974357
227 2670628
228 4723290
229 2574703
230 2635435
231 400027
232 2164430
233 1248642
234 6283358
235 6958413
236 3311417
237 8351255
238 1907630
239 8654928
240 924093
241 2998772
242 141725
243 1091750
244 3703248
245 749728
246 7232382
247 3165130
248 4443460
249 6222538
250 2463633
251 5315012
252 478160
253 4792547
254 4691848
255 811652
256 6088402
257 3515755
258 7532130
259 6281243
260 2195720
261 6950247
262 6548558
263 3503405
264 4290753
265 5494620
266 4225632
267 3418433
268 1974957
269 6039238
270 2331423
271 6206702
272 9813760
273 5335755
274 2450158
275 3031595
276 12364972
277 12908728
278 5582992
279 5159873
280 3796865
281 4553042
282 2046828
283 5461880
284 12514908
285 4172835
286 7935177
287 1653998
288 3934220
289 20158288
290 4158088
291 4820767
292 4105178
293 11322162
294 9113008
295 4135918
296 18194227
297 5011358
298 3960080
299 3129508
300 5232260
301 12167097
302 6988970
303 8117422
304 11951468
305 8106710
306 2681627
307 10473508
308 10340447
309 3210918
310 7666407
311 4619167
312 8278058
313 6400455
314 17187033
315 4702603
316 15209862
317 2442753
318 16896975
319 9072458
320 5301938
321 34220632
322 7522380
323 16858160
324 23102243
325 8320368
326 26359697
327 8442298
328 19084790
329 18287903
330 20969477
331 3583562
332 11808325
333 7186347
334 14821678
335 9228138
336 13069702
337 17811398
338 11018355
339 13511103
340 21878120
341 7581012
342 11702668
343 28398507
344 24668528
345 12763128
346 4961502
347 20779808
348 15215265
349 23685518
350 13793112
351 32928002
352 6521160
353 10293222
354 22956843
355 6302263
356 12593857
357 19960265
358 12270310
359 32898623
360 10196200
361 45449332
362 17042403
363 16624200
364 22301048
365 2897080
366 40950712
367 33111365
368 8149760
369 51585693
370 22896738
371 15476102
372 32159640
373 13446737
374 27670643
375 26215250
376 5125372
377 18111105
378 11993790
379 22279238
380 12127783
381 35827397
382 11787130
383 31053705
384 37896723
385 7089100
386 53895292
387 8877060
388 32429742
389 33247093
390 37801353
391 45403782
392 53265118
393 33932187
394 54592048
395 52420050
396 43802472
397 18049278
398 8268992
399 17798573
400 39051617
401 8438437
402 22831953
403 24129945
404 15630403
405 13404100
406 16157852
407 31911968
408 16036020
409 131974248
410 31893540
411 22425307
412 29484010
413 47602420
414 82119803
415 19775268
416 29386417
417 68602238
418 41778520
419 93495743
420 23842290
421 142202692
422 25123605
423 78731275
424 50655253
425 13855742
426 109107422
427 15573180
428 31192690
429 38107018
430 25165095
431 18371552
432 40731705
433 13834285
434 53527358
435 24080842
436 16710662
437 48520285
438 24629435
439 13132343
440 32978397
441 66760127
442 39234520
443 18875645
444 78330303
445 12612057
446 143010737
447 12373485
448 21431515
449 110247748
450 53824503
451 87678192
452 27733970
453 50006577
454 80319873
455 44847303
456 74767472
457 104140993
458 91272780
459 92708173
460 38535163
461 42691577
462 62031975
463 59148072
464 165948043
465 135221072
466 152264252
467 80953535
468 26303177
469 125150583
470 32932580
471 126663322
472 5470395
473 150927600
474 43090618
475 97938530
476 40838817
477 75363760
478 16149007
479 284804898
480 80309185
481 115470182
482 45943663
483 36014125
484 163951638
485 39471535
486 48453332
487 140964675
488 150869192
489 189444078
490 110206745
491 100964152
492 79923665
493 48026762
494 101611758
495 45163568
496 55908142
497 76410610
498 130733067
499 33934143
500 41429537
501 70253167
502 22713905
503 191574345
504 146498963
505 46437737
506 173190797
507 164960630
508 216902525
509 102749973
510 152688238
511 209690502
512 48815645
513 157862995
514 386570573
515 48852517
516 65749927
517 39161155
518 168549545
519 141866148
520 50182800
521 362098837
522 120656933
523 105170832
524 178905408
525 159979208
526 270179982
527 120056013
528 253317012
529 119288608
530 41440173
531 79966927
532 141871385
533 197378050
534 117206738
535 209633148
536 230621097
537 225768958
538 149105470
539 107503958
540 247994175
541 221250442
542 120739000
543 254478770
544 371148278
545 150743317
546 158681687
547 304231638
548 210201245
549 348848868
550 119082393
551 213351177
552 238235035
553 146988425
554 135184668
555 48778543
556 266472882
557 179175435
558 179486832
559 373649708
560 118767623
561 418105097
562 133244403
563 99186192
564 333303593
565 101807010
566 330170307
567 82334390
568 171665365
569 461913048
570 343740415
571 361007092
572 83969473
573 157917532
574 975992038
575 208413355
576 465571062
577 385601498
578 232052730
579 158837623
580 42658325
581 272912052
582 485700583
583 140274062
584 532220733
585 276236212
586 805665957
587 176958073
588 391344047
589 362709253
590 343599443
591 96230787
592 135656225
593 105832680
594 123912008
595 204503943
596 269491217
597 305953818
598 406283570
599 752414318
600 237903498
601 213329912
602 390995748
603 140998195
604 954806573
605 370473637
606 663391382
607 149046508
608 352775105
609 249857878
610 208120502
611 1000855662
612 417634343
613 552035622
614 345036853
615 242905400
616 141326852
617 810010058
618 688137917
619 271236933
620 193774213
621 136370567
622 257638638
623 319521510
624 250962968
625 559311310
626 1526295402
627 222255400
628 243023212
629 1207752558
630 173214632
631 985749397
632 337902708
633 250537490
634 65136288
635 404928923
636 538804067
637 178339938
638 275472495
639 696731203
640 421723087
641 1205491327
642 333242723
643 375536122
644 891125008
645 172730940
646 448706972
647 653288125
648 249183457
649 968771613
650 752190862
651 531453597
652 176380515
653 800990200
654 206795523
655 927234147
656 713470662
657 515432508
658 514294065
659 870602408
660 331622147
661 1020206457
662 171615535
663 455841750
664 931278113
665 282435977
666 2138760872
667 303122703
668 1051665225
669 897095818
670 929629507
671 373712582
672 571641690
673 521519827
674 1138708058
675 502328663
676 1058510057
677 1268768065
678 1185624415
679 445660388
680 393109768
681 672667072
682 440687683
683 856795660
684 1066747168
685 129325277
686 362174137
687 457057360
688 723727830
689 503095553
690 463062365
691 635356752
692 479815865
693 241485305
694 1117006223
695 379544597
696 1413201342
697 737908783
698 158264787
699 2952344623
700 266342372
701 266932207
702 518974398
703 1099292532
704 224136528
705 1284198608
706 879579992
707 837723670
708 163229195
709 255870083
710 117198610
711 981031987
712 473975068
713 252670687
714 518521913
715 281771192
716 763730312
717 1194307485
718 457563965
719 768783148
720 1430679030
721 2981704352
722 1075227960
723 645559600
724 1092654273
725 1275518800
726 405648712
727 1526926895
728 1257293142
729 1113658718
730 330027455
731 1542800172
732 1876838155
733 668620160
734 1435710433
735 758035187
736 460628752
737 1472270095
738 990524012
739 2274128533
740 666891360
741 798693837
742 528848913
743 1626095195
744 2027064553
745 1488413150
746 3024412342
747 1120822953
748 1119254007
749 1829263013
750 309474498
751 2141857912
752 1188145445
753 1555699295
754 1575788473
755 490061507
756 1499065927
757 1430876408
758 1093950312
759 658936968
760 2848292893
761 965800582
762 2794651450
763 1303968270
764 1816405818
765 316632998
766 5053512067
767 209455865
768 1026448215
769 2090858998
770 1127288990
771 3219013352
772 914895105
773 775537060
774 3098404998
775 1771212977
776 3371708917
777 982763870
778 2485321380
779 1209619483
780 517354317
781 2351990732
782 2284994583
783 3058180570
784 2332723393
785 814993582
786 2710043882
787 1602575968
788 1099506422
789 3895859993
790 2275842880
791 3801465167
792 1212027450
793 1109824522
794 1596204108
795 116423748
796 1311811142
797 3826190170
798 1838108060
799 2075033258
800 2590674053
801 3991537872
802 1584876305
803 688762370
804 2907115753
805 1963865683
806 1218082057
807 2783167338
808 1244246607
809 6688914953
810 1136057325
811 4879530092
812 1078187868
813 1963011935
814 2456814678
815 817898587
816 2105301987
817 1476418680
818 1178478367
819 2422738013
820 1150105098
821 2446124457
822 2962621393
823 854904790
824 3118041113
825 1596971260
826 1545818592
827 1516269855
828 1080542262
829 1468003588
830 2194392940
831 7926904447
832 3623219465
833 2053566580
834 1701857453
835 1738411727
836 521604192
837 2851193245
838 707757395
839 8054272678
840 1164979197
841 3999978077
842 1559932698
843 4593483547
844 4196367348
845 1449938513
846 3627224892
847 2271177758
848 5492863497
849 6230400578
850 1203578523
851 3810733792
852 2575036223
853 6060960015
854 1931713668
855 1819220120
856 3172774052
857 1482890350
858 4818795527
859 5689829723
860 6177395698
861 2405795462
862 3026856388
863 2405159977
864 11106595143
865 4289287007
866 4209447767
867 3849096298
868 5069409247
869 7166434928
870 2064869648
871 2476926237
872 4687880825
873 4000930010
874 4776478178
875 995769922
876 1233671862
877 836299100
878 1323609910
879 9304370708
880 2311285048
881 4678670617
882 411106845
883 3147293557
884 809445493
885 2306786122
886 3287184092
887 1031407750
888 5187146730
889 8226414758
890 3048037043
891 4781139942
892 4455278398
893 5321666817
894 4568127788
895 6559094538
896 3352338792
897 4322805470
898 2179079075
899 2248681003
900 2238954123
901 4346895982
902 7060275863
903 1223175910
904 11060909463
905 1574161108
906 4349096642
907 2526316303
908 5763562942
909 9224381248
910 3187343453
911 4162545082
912 3920873523
913 6284266240
914 7887840663
915 3964391842
916 4023472667
917 4967220493
918 2785190895
919 4516769603
920 1654347340
921 6886816167
922 1522788600
923 1731506367
924 4163553853
925 1507820423
926 6972833117
927 2528311860
928 6198848495
929 5310531548
930 2689512325
931 2719152782
932 6172499860
933 4207178020
934 3005762743
935 4422668477
936 18462023822
937 2615806835
938 4397130915
939 8493754803
940 1299446533
941 8527231937
942 4945354223
943 3912129900
944 13643349213
945 5531673495
946 11543632172
947 4015179735
948 12862817032
949 9407061128
950 5657024718
951 3749254647
952 7884111378
953 3780605152
954 4441086328
955 9549957132
956 9626553492
957 3765668163
958 5365459852
959 14906845753
960 4077390530
961 856359122
962 8640331193
963 1863989655
964 10792231053
965 3157455605
966 6129872887
967 9973337810
968 5706917330
969 15473299583
970 3670338742
971 7514667242
972 14374380998
973 5016435072
974 17508006308
975 1065472893
976 4204219947
977 16162465793
978 8987211702
979 9417392003
980 6262281762
981 13519426337
982 10207490848
983 11779435180
984 16686552823
985 13121206348
986 5917519772
987 7103990803
988 3770789082
989 10920336278
990 6159537563
991 3553869007
992 7971882450
993 16539204720
994 3923891423
995 11323476390
996 2829945767
997 8108352970
998 15359688132
999 18263276708
1000 4718578212
1001 2811629627
1002 11024680285
1003 6835356575
1004 19374221148
1005 714897587
1006 14257463852
1007 4267983675
1008 10764699650
1009 18724975378
1010 4959775740
1011 6904565602
1012 6009418280
1013 6582996530
1014 11991672908
1015 7504060620
1016 12499155547
1017 8955975688
1018 5986929792
1019 7242983443
1020 13688528333
1021 12937575437
1022 6003407428
1023 8561645130
[/CODE]

rudy235 2019-04-16 23:22

[QUOTE=ATH;513889]First occurrence gaps up to gap=1023.


[CODE]
gap k (before gap)
1 1
2 3
3 7
4 103
5 12
6 52
7 378
8 87
9 313
10 77
11 287
12 58
13 192
14 298
15 597
16 357
17 1075
18 220
19 [COLOR="red"]3563[/COLOR]
20 248
21 2042
22 800
23 147
24 [COLOR="red"]3843[/COLOR]
25 110
26 [color="red"]3257[/COLOR]
27 2063
28 397
29 [COLOR="Red"]6458[/COLOR]
30 1755



1000 4718578212
1001 2811629627
1002 11024680285
1003 6835356575
1004 19374221148
1005 714897587
1006 14257463852
1007 4267983675
1008 10764699650
1009 18724975378
1010 4959775740
1011 6904565602
1012 6009418280
1013 6582996530
1014 11991672908
1015 7504060620
1016 12499155547
1017 8955975688
1018 5986929792
1019 7242983443
1020 13688528333
1021 12937575437
1022 6003407428
1023 8561645130
[/CODE][/QUOTE]

Thank very much ATH for that.

I could only modestly go until k≤ 2700 because I was doing it in EXCEL.

I could find all the first occurrences from 1-18 and the next few ones up to 30 with the exception of 19, 24, 26 and 29

Seems clear from what you have done that it can be safely assumed (but of course not easy to prove) that ALL gaps are going to be present.

retina 2019-04-17 01:03

[QUOTE=rudy235;513891]... I was doing it in EXCEL.[/QUOTE]If you are serious about doing this kind of stuff then download and learn to use some dedicated computing software.

ATH 2019-04-17 01:09

Continued up to k=200B. Here are consecutive first occurrence gaps 1024 up to 1358, as well as the other nonconsecutive first occurrence gaps 1360 to 1898.

Also added the maximal gap list in order of occurrence.

[CODE]gap k
1024 30370627668
1025 16507782197
1026 12133392337
1027 8527092578
1028 5400391645
1029 8780665563
1030 5007391888
1031 8309499172
1032 3740360183
1033 12579042990
1034 7490905648
1035 4354724050
1036 12790338547
1037 14585762935
1038 12918899037
1039 10528753153
1040 8643398725
1041 22228983707
1042 18421312110
1043 4277863287
1044 22247337703
1045 9117393532
1046 11812293782
1047 3196956045
1048 7919321767
1049 10991916183
1050 9865733768
1051 22563333397
1052 22070444965
1053 16686868527
1054 12996465758
1055 8472371157
1056 31366197567
1057 9491799368
1058 8659623317
1059 11055610988
1060 13235724187
1061 41915068062
1062 18498428755
1063 9023952725
1064 9653598588
1065 5298861998
1066 12036778707
1067 14334530700
1068 39987891870
1069 22736738038
1070 4416179072
1071 12539957742
1072 25946320698
1073 14636179420
1074 12432480398
1075 7762655897
1076 19372935412
1077 14255597088
1078 16521687485
1079 4035849583
1080 24070972425
1081 21562226522
1082 14943718163
1083 13624623332
1084 18383401348
1085 15837017163
1086 43177754272
1087 18253515870
1088 17472321867
1089 11557132158
1090 8186246962
1091 41500930727
1092 4142929803
1093 9490473660
1094 21439776823
1095 40213458707
1096 28895725292
1097 21386758558
1098 25137783535
1099 29311613548
1100 14524563518
1101 12875454587
1102 10942894720
1103 11202556685
1104 16615629933
1105 16606921495
1106 20860243742
1107 17500043025
1108 6708903010
1109 19821577948
1110 12852163973
1111 8412207522
1112 25186770675
1113 14333623160
1114 23078677218
1115 9254604950
1116 15134152567
1117 24189828365
1118 29646936667
1119 34226827423
1120 18463125062
1121 58663327587
1122 10133636193
1123 26227925140
1124 42383800163
1125 38347022838
1126 20497482947
1127 8185647088
1128 29451224835
1129 29579425773
1130 16298537952
1131 37485937562
1132 7578097340
1133 25913710302
1134 14524565688
1135 20792431795
1136 28809229977
1137 13582827433
1138 37975916165
1139 61756317343
1140 15892671560
1141 10439154257
1142 25262591533
1143 28593081655
1144 48826150303
1145 41705622962
1146 25794434892
1147 16293399008
1148 8582270267
1149 8525721438
1150 16251349042
1151 45316370542
1152 21034892753
1153 15728686990
1154 17504175188
1155 24735441112
1156 33174606362
1157 27616548440
1158 42035688247
1159 23092251588
1160 14998583343
1161 56801711312
1162 29812173248
1163 17756220570
1164 16797436063
1165 14435182932
1166 15804331282
1167 33766997653
1168 29664641207
1169 51537427428
1170 25320667690
1171 21420242172
1172 39668112358
1173 13791132407
1174 18761657808
1175 7073619248
1176 30537616247
1177 23153721783
1178 15554271252
1179 57841170578
1180 13255405793
1181 31234475137
1182 30924553110
1183 26168680170
1184 49319201213
1185 23812067153
1186 41936084087
1187 8939732550
1188 14330487105
1189 44674906188
1190 8031358767
1191 10603108182
1192 18835412365
1193 34746153190
1194 16560031848
1195 9391637273
1196 14528709892
1197 18242511845
1198 25046740252
1199 33650418018
1200 18359485235
1201 33051842117
1202 66002847468
1203 26782423792
1204 40847394758
1205 15928146832
1206 64999749057
1207 9988457248
1208 19986121662
1209 53500919883
1210 29408935490
1211 32056558362
1212 46819768958
1213 23359512832
1214 20331890558
1215 36293533640
1216 68006592497
1217 48101859580
1218 35070286585
1219 10774224213
1220 36726742647
1221 25678337692
1222 8695593693
1223 43949266767
1224 90705803338
1225 25905710483
1226 35530520182
1227 50758082210
1228 43848563047
1229 44351452008
1230 26404449530
1231 101208274662
1232 22556325743
1233 37269319510
1234 45484451658
1235 41601982865
1236 20875795512
1237 56829323030
1238 32867985517
1239 33904468903
1240 37951687547
1241 23524711222
1242 51497620853
1243 64872205377
1244 65789117053
1245 57057845338
1246 40884738972
1247 36050767643
1248 53024192545
1249 48725636383
1250 28543693757
1251 64133691577
1252 45976631095
1253 56015688972
1254 100761544543
1255 22544518132
1256 38237309287
1257 37685943368
1258 20020326737
1259 89817312058
1260 40437430637
1261 81952221312
1262 34386364680
1263 64312175652
1264 41031871478
1265 7595970438
1266 66001896202
1267 102711625448
1268 52055075942
1269 60464331573
1270 26072298478
1271 94947179337
1272 37482239365
1273 52338755872
1274 31605073873
1275 37883577230
1276 19485934097
1277 10846539718
1278 53318021160
1279 80634404163
1280 40551655827
1281 31260613627
1282 51115333138
1283 69732506812
1284 70023017393
1285 12347431502
1286 41906763817
1287 29873248415
1288 29314359620
1289 44402436488
1290 43944263470
1291 60328695492
1292 35124127965
1293 44978931772
1294 78981727158
1295 66748779720
1296 136174101612
1297 83379921555
1298 9634521157
1299 29706714558
1300 59173429653
1301 64935700882
1302 77686997383
1303 68044441195
1304 40831407493
1305 24030385738
1306 37042953262
1307 92308689725
1308 91890048212
1309 69089723273
1310 71131302125
1311 65562906117
1312 50485273458
1313 48624178687
1314 83852500748
1315 48882377632
1316 75146429732
1317 36692181453
1318 53992589912
1319 186438133443
1320 51173468143
1321 30530362357
1322 94707449170
1323 42450011142
1324 70051577718
1325 40320006480
1326 78720103417
1327 31520502065
1328 34445495270
1329 143446677043
1330 7287610437
1331 86046387072
1332 51772611128
1333 104162957980
1334 44438479298
1335 46412332407
1336 92479857902
1337 51780274385
1338 97017192822
1339 79715880988
1340 10849288625
1341 82083852107
1342 132223116333
1343 25083224815
1344 62994611978
1345 78773730823
1346 103895222522
1347 66824446040
1348 93470563142
1349 46032970723
1350 110940120422
1351 147555423337
1352 44028349955
1353 72388630452
1354 176010608498
1355 35415187398
1356 73721641492
1357 39997802863
1358 68875590515



1360 120701590522
1361 103656076897
1362 129780127055
1363 45464706810
1365 27432631065
1367 66664704428
1368 79675959260
1369 105795508798
1370 22751684835
1371 150422884902
1372 73987959848
1373 91101657665
1374 74619205678
1375 106433204497
1376 109109662112
1377 140926519020
1378 158222902962
1379 59513006413
1380 83535481318
1381 81120569787
1383 82037154155
1384 91105669713
1385 35954269187
1386 36654125322
1387 82436707565
1388 52106714982
1390 134783026287
1391 91527303572
1392 160241217295
1393 78441568602
1394 107218832948
1395 93204583073
1397 71529209675
1398 89717150497
1399 167763335213
1400 78769799252
1401 113406055667
1402 83591984703
1404 115946907303
1405 60066856568
1406 157757582267
1408 75688827280
1409 131711929323
1410 199574686315
1411 48278160327
1412 157757709578
1413 32135929095
1414 179075794238
1415 49037845735
1416 64167955752
1417 117577979190
1418 108513597027
1419 77078300443
1420 74951308542
1421 42119304362
1422 101735004478
1423 142152878497
1425 88942037140
1426 114261540757
1428 100389566412
1429 175835907578
1430 48978765967
1431 105371512887
1432 89851341625
1434 197709087693
1435 110562730142
1436 34440671017
1437 141185526520
1438 167850568830
1439 94595234198
1440 34516209527
1441 195969981917
1442 136888972125
1443 134581666552
1444 159728123338
1445 38840382823
1447 57301290370
1448 77920988750
1449 102332614223
1450 108516370937
1451 64095115322
1452 151527991675
1453 145272707550
1454 174799142838
1455 81631078743
1456 195778814802
1460 64584093742
1462 88475830910
1463 69741598787
1465 135767046975
1467 85661945075
1468 63139736062
1470 106208976653
1471 156171984272
1472 138299223690
1473 184783689402
1475 67410836280
1477 185821439738
1478 96139203857
1482 176427443750
1483 48773094382
1484 136319467498
1485 169809420105
1486 150516739082
1489 130741136258
1490 88340661855
1492 121639858400
1493 66924164457
1495 69782047618
1496 174973906777
1499 22339570278
1500 172823544343
1502 128837925765
1503 181674927455
1505 160823433270
1507 176020578323
1510 163775464003
1515 155390806498
1516 85283577452
1517 196413282468
1518 184159612907
1520 159460255028
1523 147488057610
1525 98616660978
1526 56105289977
1527 194013931730
1529 137914593968
1530 164942161335
1531 82643395422
1536 151913191462
1538 119496454527
1540 139666887840
1542 181392272663
1543 180056667807
1544 146578135928
1545 70464152903
1546 168996745347
1547 160429275275
1548 142976782372
1549 170014476063
1550 174553795390
1552 33051947625
1553 37182176622
1555 119208146283
1556 107282165062
1557 143282458305
1558 164872223950
1560 62578713270
1561 176158811742
1562 118027566675
1565 136243056022
1575 103750795712
1577 185624758875
1580 184298726117
1583 129919858890
1585 96643867775
1587 144666194755
1590 116320889387
1595 177702480225
1603 121584681445
1605 185545828858
1610 109354338662
1613 128518326722
1618 151521188780
1620 111423209912
1631 117959421867
1633 103293294022
1635 101323871015
1643 158850706422
1646 197557993677
1649 158176828393
1658 166226961612
1659 165136905488
1660 133864375453
1664 131415761768
1672 161263436148
1687 156290145345
1700 58917239540
1710 89470349442
1723 80799633875
1778 106405396032
1785 130744752537
1810 133135498198
1815 169866686215
1890 151287342070
1898 132437316545[/CODE]

[CODE]maxgap k
1 1
2 3
3 7
5 12
6 52
12 58
25 110
28 397
35 980
47 2233
62 3090
83 4070
105 10383
154 31318
155 114587
168 114742
242 141725
252 478160
255 811652
287 1653998
317 2442753
365 2897080
376 5125372
472 5470395
478 16149007
502 22713905
517 39161155
530 41440173
580 42658325
634 65136288
795 116423748
882 411106845
1005 714897587
1047 3196956045
1079 4035849583
1092 4142929803
1108 6708903010
1175 7073619248
1330 7287610437
1340 10849288625
1499 22339570278
1552 33051947625
1553 37182176622
1700 58917239540
1723 80799633875
1778 106405396032
1785 130744752537
1898 132437316545
[/CODE]

robert44444uk 2019-04-17 08:21

These gaps do look interesting.

If ATH shared his program, we could co-ordinate this to higher levels.

The values of k of the first instance gaps are much smaller than those for prime gaps, even when the latter are divided by two.

I wonder whether it is possible to hack Robert G.'s prime gap program to get a massive speed up?

Thomas11 2019-04-17 13:53

1 Attachment(s)
A quick and easy solution would be using NewPGen (sieving for twin primes of the type k*6^1+/-1) and then using some Perl script or similar tool to generate and update the gap list.

In a Linux environment the following bash script would do the job:

[CODE]kmin=1
kmax=100010000
kstep=100000000
kfinal=1000010000

while [ $kmax -le $kfinal ]
do
./newpgen -v -t=2 -base=6 -n=1 -kmin=$kmin -kmax=$kmax -wp=6k.txt
./update_gaplist.pl 6k.txt
kmin=`expr $kmin + $kstep`
kmax=`expr $kmax + $kstep`
done
[/CODE]

The necessary Perl script can be found in the attached ZIP file.

The range up to k=1B takes less than a minute. But note that the sieve time increases with increasing k (roughly with the square root of k).

(We use a little overlap between consecutive ranges in order to make sure that we do not loose any gaps between the intervals.)

Dr Sardonicus 2019-04-17 17:59

Earlier, this Forum was treated to a link about [url=https://projecteuclid.org/euclid.em/1047477055]Jumping champions[/url], having to do with the differences between consecutive primes. Using the table of the first 100k (not 10k) twin primes (I dropped the "3" and just looked at the 99999 remaining ones) I found that if p = 6*k - 1 and p' = 6*k' - 1 are consecutive, then

the largest value of g = k' - k which occurs is 365.

I decided to compile a list of gap values g = k' - k from g = 1 to g = 365. I list the first 65 of these. It looks like, at this stage, g = 5 is somewhat favored [pairs of twin primes differing by 6*5 = 30], with g = 7 a close second and g = 2 a somewhat distant third.

[1377, 3482, 2465, 1660, 4804, 1386, 4342, 2368, 1488, 3352, 1637, 2608, 2476, 1630, 3195, 1272, 2007, 1968, 1121, 2345, 1283, 2026, 2183, 548, 2214, 1201, 1281, 2087, 658, 2459, 579, 1170, 1670, 616, 2545, 409, 1359, 969, 520, 1872, 372, 1152, 732, 592, 988, 333, 773, 774, 467, 798, 505, 578, 519, 290, 920, 370, 460, 606, 197, 595, 329, 536, 611, 149, 807]

If someone is aware of "jumping champions" heuristics for twin primes, please share.

rudy235 2019-04-17 18:27

[QUOTE=robert44444uk;513934]These gaps do look interesting.

The values of k of the first instance gaps are much smaller than those for prime gaps, even when the latter are divided by two.

[/QUOTE]

Yes, Robert I did notice that. While it took years and years of search to get to the first kilogap by B. Nyman in 2001 (in case of the primes) which is really a gap of 566 if divided by 2, in this case, it was relatively easy for ATH to get to 1023.

Just to add one other fact. As the gaps of 1 are _by definition_ Quadruplet Primes, we can make an accurate estimation of how many gaps of 1 they are for a particular level.


So for instance π (10[SUP]12[/SUP]) is 37,607,912,018 while π_4(10[SUP]12[/SUP]) is 8,398,278
In other words at that level (10[SUP]12[/SUP]) [B]for every 4478[/B] gaps [B]one of those is equal to 1 [/B]. At the 10[SUP]16[/SUP] level the ratio goes down to [B]1 for every 11,002 gaps[/B]

T. R . Nicely has tabulated the Quad primes up to 1.p*10[SUP]16[/SUP] [URL="http://www.trnicely.net/quads/quads.html"][SIZE="3"]here[/SIZE][/URL]
[code]
x π_[SUB]4 [/SUB](x) π(x)
======================
1e07 899 664,579
1e08 4768 5,761,455
1e09 28388 50,847,534
1e10 180529 455,052,511
1e11 1209318 4,118,054,813
1e12 8398278 37,607,912,018
1e13 60070590 346,065,536,839
1e14 441296836 3,204,941,750,802
2e14 807947960 6,270,424,651,315
3e14 1151928827 9,287,441,600,280
4e14 1482125418 12,273,824,155,491
5e14 1802539207 15,237,833,654,620
6e14 2115416076 18,184,255,291,570
7e14 2422194981 21,116,208,911,023
8e14 2723839871 24,035,890,368,161
9e14 3021126140 26,944,926,466,221
1e15 3314576487 29,844,570,422,669
1.1e15 3604646822 32,735,816,605,908
1.2e15 3891706125 35,619,471,693,548
1.3e15 4175985018 38,496,205,973,965
1.4e15 4457782901 41,366,582,391,891
1.5e15 4737286827 44,231,080,178,273
1.6e15 5014641832 47,090,114,439,072
1.7e15 5290057283 49,944,045,778,207
1.8e15 5563600032 52,793,190,012,734
1.9e15 5835422569 55,637,829,945,151
2.0e15 6105617289 58,478,215,681,891
3.0e15 8734892736 86,688,602,810,119
4.0e15 11265509044 114,630,988,904,000
5.0e15 13725978764 142,377,417,196,364
6.0e15 16132120984 169,969,662,554,551
7.0e15 18494314750 197,434,994,078,331
8.0e15 20819642284 224,792,606,318,600
9.0e15 23113346779 252,056,733,453,928
1.0e16 25379433651 279,238,341,033,925
[/code]

rudy235 2019-04-18 00:42

CORRECTION:

T. R . Nicely has tabulated the Quad primes up to 1*10[SUP]16[/SUP] [URL="http://www.trnicely.net/quads/t4_0000.htm"][SIZE="3"]here[/SIZE][/URL]


All times are UTC. The time now is 05:33.

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.