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 2019-04-16, 18:20 #1 rudy235     Jun 2015 Vallejo, CA/. 2·17·29 Posts 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} … A077800 Chris Caldwell has a link to the first 10k Twin primes first element of twin primes 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 A002822 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 +/-1 The first gap of 1 appears at the start as we can see here Code: k Gap 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
 2019-04-16, 19:01 #2 rudy235     Jun 2015 Vallejo, CA/. 3DA16 Posts 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 are theorized to be infinite 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 1 (first time) 2 1 3 2 (first time) 5 2 7 3 (first time) 10 2 12 5 (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 6 (first time) 58 12 (first time) 70 2 72 5 77 10 (first time) 87 8 (first time) 95 5 100 3 103 4 (first time) 107 3 110 25 (first time) 135 2 137 1 138 5 143 4 147 23 (first time) 170 2 172 3 175 2 177 5 182 10 192 13 (first time) 205 8 213 2 215 2 217 3 220 18 (first time) 238 4 242 5 247 1 248 20 (first time) 268 2 270 8 278 5 283 4 287 11 (first time) 298 14 (first time) 312 1 313 9 (first time) 322 3 325 8 333 5 338
 2019-04-16, 22:29 #3 ATH Einyen     Dec 2003 Denmark C3F16 Posts 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 Last fiddled with by ATH on 2019-04-16 at 22:32
2019-04-16, 23:22   #4
rudy235

Jun 2015
Vallejo, CA/.

2·17·29 Posts

Quote:
 Originally Posted by ATH 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 … … 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
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.

Last fiddled with by rudy235 on 2019-04-17 at 00:14 Reason: added color

2019-04-17, 01:03   #5
retina
Undefined

"The unspeakable one"
Jun 2006
My evil lair

6,143 Posts

Quote:
 Originally Posted by rudy235 ... I was doing it in EXCEL.
If you are serious about doing this kind of stuff then download and learn to use some dedicated computing software.

 2019-04-17, 01:09 #6 ATH Einyen     Dec 2003 Denmark C3F16 Posts 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 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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: 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
 2019-04-17, 08:21 #7 robert44444uk     Jun 2003 Oxford, UK 1,933 Posts 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? Last fiddled with by robert44444uk on 2019-04-17 at 08:21
2019-04-17, 13:53   #8
Thomas11

Feb 2003

35648 Posts

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
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.)
Attached Files
 twingaps.zip (822 Bytes, 202 views)

Last fiddled with by Thomas11 on 2019-04-17 at 14:00

 2019-04-17, 17:59 #9 Dr Sardonicus     Feb 2017 Nowhere 23·569 Posts Earlier, this Forum was treated to a link about Jumping champions, 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. Last fiddled with by Dr Sardonicus on 2019-04-17 at 18:00 Reason: finxig spoty
2019-04-17, 18:27   #10
rudy235

Jun 2015
Vallejo, CA/.

98610 Posts

Quote:
 Originally Posted by robert44444uk 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. …
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 (1012) is 37,607,912,018 while π_4(1012) is 8,398,278
In other words at that level (1012) for every 4478 gaps one of those is equal to 1 . At the 1016 level the ratio goes down to 1 for every 11,002 gaps

T. R . Nicely has tabulated the Quad primes up to 1.p*1016 here
Code:
     x         π_4 (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

 2019-04-18, 00:42 #11 rudy235     Jun 2015 Vallejo, CA/. 17328 Posts CORRECTION: T. R . Nicely has tabulated the Quad primes up to 1*1016 here

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