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[QUOTE=robert44444uk;126329]Here is a table of lowest k for each twin to n=1000
Does anyone want to take further? * denotes jumping champion [/QUOTE] hi robert, i hope you spent not much time in that! please have a look here: [url]http://www.rieselprime.org/FirstKTwin.htm[/url]. i made this page 4 months ago after i found a link on this first twin for a k (see page). all n upto 1130 are filled in there and with gaps upto 1400. i want to expand this page but found no time yet! karsten |
[QUOTE=kar_bon;126450]hi robert,
i hope you spent not much time in that! please have a look here: [url]http://www.rieselprime.org/FirstKTwin.htm[/url]. i made this page 4 months ago after i found a link on this first twin for a k (see page). all n upto 1130 are filled in there and with gaps upto 1400. i want to expand this page but found no time yet! karsten[/QUOTE] No only spent about an hour or so fiddling about. I think it is an interesting exercise because of the statistics of series like this. If you plot all the values then it increases gently and there is a nice curve bounding 95% of values, but the outer envelope of the rogue 5% seem to be increasing on a much steeper curve. I have a hunch that you could look at really rather a large set of consecutive n's then you would be certain to find a twin prime over a relatively small k range, and that would be more efficient that the current k*2^333333+/-1 twin search, as 333333 might be just such a rogue!! Statisticians could I think define such a range of manageable k and n to produce 99% probability of getting a twin. Whereas n=333333 - who knows! |
[quote=robert44444uk;126813]No only spent about an hour or so fiddling about.
I think it is an interesting exercise because of the statistics of series like this. If you plot all the values then it increases gently and there is a nice curve bounding 95% of values, but the outer envelope of the rogue 5% seem to be increasing on a much steeper curve. I have a hunch that you could look at really rather a large set of consecutive n's then you would be certain to find a twin prime over a relatively small k range, and that would be more efficient that the current k*2^333333+/-1 twin search, as 333333 might be just such a rogue!! Statisticians could I think define such a range of manageable k and n to produce 99% probability of getting a twin. Whereas n=333333 - who knows![/quote] I agree completely Robert. I suggested this to MooMoo, the TPS project leader, in an Email several weeks ago. I didn't hear anything back. I think they've already decided that they'll do a fixed-n search on n=500K after they find a twin for n=333333. IMHO, searching a moderate-sized range of consecutive n over a much smaller range of k-values is more effecient in the long run due to the LLRing efficiency gained from the smaller k-values. In theory, the chances are the same either way of accidently searching a bad range. The TPS way, it's a 'rogue' n-value. Using the way I did it, you'd have the same number of candidates with more n-values but less k-values but still the same chance of catching a rogue k/n range. But the reason why I think it is more efficient is because smaller k's LLR so much faster. Even though you lose sieving depth and efficiency, LLRing is a large percentage of any prime search so more effort should be given towards minimizing LLR time instead of sieve time. To get to where I've searched so far on my effort, it's only taken a few months of 2 pretty slow older 2.66 Ghz P4's. If I put 10-12 faster CPU's on it, I could most likely search n=36K-100K faster than I did just getting to n=36K. Gary |
A simple exercise to prove the above.
Take the first instance k for the first 1,000 n, shown in the table posted above on 21 February. Take all combinations of 15 consecutive n, and calulate the average of the k values chosen (=A), and the minimum k in the range. Multiply the minimum by 15 (=B), and compare to the average. A is smaller than B in 197 cases, but B is smaller than A in 788 cases. The broader the range so the number of cases that B<A increases: range of n A/B B<A as % of total cases 10 371/619 62.5% 15 197/788 80.0% 20 109/871 88.9% 25 74/901 92.4% 30 54/916 94.4% So I really do not understand the 333333 search!!!! |
[quote=robert44444uk;126329]Here is a table of lowest k for each twin to n=1000
Does anyone want to take further? * denotes jumping champion [code] 1 3* 2 1 3 9* 4 15* 5 81* 6 3 7 9 8 57 9 45 10 15 11 99* 12 165* 13 369* 14 45 15 345 16 117 17 381* 18 3 19 69 20 447* 21 81 22 33 23 1179* 24 243 25 765 26 375 27 81 28 387 29 45 30 345 31 681 32 585 33 375 34 267 35 741 36 213 37 429 38 3093* 39 165 40 267 41 255 42 1095 43 9 44 147 45 849 46 405 47 1491 48 177 49 1941 50 927 51 1125 52 1197 53 2001 54 333 55 519 56 1065 57 585 58 657 59 129 60 147 61 141 62 417 63 9 64 1623 65 99 66 2985 67 2469 68 4497* 69 5259* 70 597 71 7029* 72 315 73 3081 74 2457 75 4161 76 603 77 3591 78 2697 79 3681 80 213 81 2079 82 1545 83 4089 84 165 85 1455 86 10287* 87 1629 88 387 89 3321 90 14487* 91 849 92 1467 93 3339 94 3747 95 6639 96 7737 97 8265 98 15735* 99 5589 100 4107 101 9225 102 537 103 2079 104 1203 105 1515 106 1323 107 7245 108 6897 109 20631* 110 2205 111 2175 112 3087 113 11145 114 7887 115 14841 116 2673 117 5961 118 3303 119 5565 120 3957 121 9849 122 1497 123 1125 124 1983 125 699 126 2565 127 8721 128 4467 129 5835 130 6063 131 1089 132 3117 133 1455 134 3105 135 6129 136 22365* 137 3555 138 24453* 139 8121 140 4143 141 1179 142 6903 143 309 144 11505 145 14121 146 17037 147 1419 148 17157 149 5715 150 345 151 13179 152 4497 153 3741 154 10803 155 105 156 30657* 157 14439 158 14445 159 7569 160 17295 161 25425 162 6555 163 2121 164 3717 165 13731 166 7737 167 18711 168 765 169 1881 170 19335 171 32361* 172 2847 173 2115 174 4155 175 1941 176 1383 177 24771 178 2277 179 10479 180 4287 181 441 182 19617 183 27261 184 2493 185 5481 186 28227 187 20175 188 1935 189 45 190 525 191 13719 192 8337 193 12495 194 18087 195 27099 196 9753 197 56745* 198 4245 199 8265 200 63855* 201 27261 202 69855* 203 14199 204 1755 205 5529 206 1197 207 54639 208 69753 209 10461 210 10575 211 9 212 3615 213 26145 214 9225 215 5859 216 12255 217 6615 218 16653 219 18531 220 24087 221 6555 222 7947 223 12909 224 49203 225 49341 226 10857 227 3405 228 25665 229 19041 230 21255 231 2571 232 30015 233 47079 234 24915 235 77751* 236 33333 237 16641 238 135 239 17289 240 10197 241 4059 242 1023 243 50319 244 22113 245 9915 246 17535 247 19041 248 15795 249 168831* 250 23007 251 5139 252 17787 253 15519 254 12957 255 1215 256 64647 257 9951 258 74253 259 2805 260 2475 261 15711 262 25767 263 9789 264 165 265 13209 266 19593 267 33105 268 45213 269 969 270 98907 271 19335 272 22317 273 10635 274 13713 275 34245 276 41085 277 24129 278 26025 279 24579 280 128505 281 3381 282 165 283 20175 284 23853 285 25881 286 61647 287 39315 288 2667 289 67695 290 34647 291 1899 292 33735 293 48861 294 2373 295 58179 296 66507 297 9609 298 20085 299 6405 300 230085* 301 44529 302 16575 303 22815 304 99297 305 21015 306 21075 307 91455 308 9993 309 15069 310 9543 311 79719 312 36195 313 14649 314 7605 315 67461 316 16035 317 12951 318 20295 319 41349 320 82473 321 20781 322 19293 323 88791 324 55605 325 23295 326 25473 327 10071 328 28653 329 48489 330 12477 331 7791 332 345675* 333 669 334 16437 335 42699 336 93765 337 12909 338 5253 339 23415 340 128625 341 21585 342 76995 343 153645 344 573 345 31719 346 15717 347 43011 348 33765 349 28149 350 71253 351 127305 352 14727 353 85431 354 10545 355 7785 356 38853 357 70851 358 65385 359 9129 360 162243 361 5049 362 49815 363 26871 364 210447 365 9369 366 74763 367 18669 368 16905 369 49299 370 12543 371 3321 372 138765 373 151839 374 40257 375 26679 376 14223 377 23709 378 22713 379 66039 380 1023 381 67749 382 34683 383 114951 384 126747 385 72609 386 114687 387 1701 388 56817 389 10791 390 39345 391 615 392 108195 393 95151 394 67023 395 21315 396 28065 397 24039 398 19065 399 102795 400 48207 401 28941 402 83337 403 101535 404 22887 405 74085 406 35253 407 79215 408 31635 409 36825 410 50835 411 273429 412 58065 413 86061 414 39513 415 17061 416 32025 417 30705 418 1743 419 71919 420 224415 421 66075 422 84057 423 81651 424 65337 425 237765 426 251475 427 83139 428 36903 429 39039 430 110157 431 66219 432 69477 433 50181 434 54033 435 5415 436 30987 437 102309 438 24693 439 56259 440 25077 441 15255 442 18795 443 3921 444 35793 445 9345 446 18663 447 30849 448 57717 449 69285 450 155463 451 26355 452 258345 453 17631 454 65193 455 2085 456 9063 457 15561 458 4323 459 104661 460 34725 461 92235 462 229227 463 53991 464 63903 465 24351 466 12147 467 33351 468 2565 469 108795 470 5547 471 139935 472 8787 473 184281 474 49053 475 13935 476 33375 477 33315 478 141315 479 53019 480 162897 481 233115 482 143163 483 150939 484 50295 485 27975 486 101055 487 156051 488 7503 489 73671 490 37095 491 37719 492 1995 493 97449 494 39207 495 27261 496 208845 497 99015 498 37755 499 131439 500 52305 501 207945 502 35397 503 66735 504 35877 505 74985 506 103107 507 5565 508 216243 509 107631 510 262035 511 43485 512 51765 513 134115 514 53355 515 87951 516 12045 517 66375 518 366555* 519 83211 520 4257 521 17709 522 80175 523 76089 524 47403 525 5775 526 62337 527 43371 528 43137 529 10365 530 74367 531 104409 532 347457 533 396441* 534 84627 535 278535 536 49893 537 23541 538 2007 539 12711 540 174297 541 8031 542 121065 543 40119 544 330015 545 18801 546 297 547 5979 548 97293 549 157209 550 26853 551 4035 552 29187 553 190485 554 70923 555 67329 556 130227 557 105381 558 80385 559 300561 560 39243 561 112581 562 176205 563 199989 564 117243 565 120069 566 75225 567 28131 568 239247 569 60411 570 25485 571 27909 572 20037 573 14259 574 70107 575 38835 576 247035 577 126615 578 136413 579 404871* 580 88257 581 76569 582 22587 583 28005 584 15177 585 210051 586 83175 587 173355 588 50235 589 133911 590 42777 591 389799 592 86385 593 45315 594 179163 595 257529 596 41625 597 268461 598 147135 599 74229 600 82023 601 135585 602 190695 603 33885 604 113475 605 264849 606 129705 607 368775 608 217143 609 228651 610 86973 611 8781 612 47313 613 94005 614 261075 615 34059 616 79353 617 29919 618 54015 619 18429 620 55203 621 46035 622 87795 623 12285 624 143265 625 104091 626 16323 627 140739 628 137907 629 223569 630 643737* 631 229749 632 506475 633 123891 634 242523 635 52419 636 78033 637 137835 638 227283 639 198459 640 558087 641 664941* 642 394203 643 91629 644 84045 645 274395 646 250923 647 24249 648 78453 649 109809 650 3723 651 205251 652 375843 653 624165 654 61353 655 38835 656 256605 657 21999 658 75447 659 101661 660 50943 661 77505 662 32067 663 374901 664 567573 665 258651 666 249345 667 127041 668 144717 669 58725 670 392013 671 130689 672 15993 673 178689 674 252693 675 376929 676 257613 677 3405 678 169893 679 469755 680 7605 681 217221 682 386127 683 151845 684 24537 685 243879 686 141705 687 246405 688 224625 689 13689 690 46545 691 38229 692 47937 693 152421 694 13197 695 2985 696 96813 697 102789 698 157587 699 436095 700 179865 701 317481 702 169827 703 5355 704 253995 705 330171 706 312387 707 37149 708 270177 709 158115 710 60693 711 25029 712 700005* 713 92529 714 35817 715 629211 716 118413 717 78561 718 86193 719 101361 720 89577 721 119721 722 150567 723 715449* 724 102213 725 20115 726 213 727 35589 728 30933 729 343359 730 308853 731 111285 732 142047 733 597339 734 50025 735 123585 736 5013 737 11175 738 95937 739 140481 740 51975 741 170625 742 58683 743 48075 744 9753 745 9165 746 131937 747 113271 748 42777 749 227871 750 22407 751 1025925 752 140967 753 110775 754 797433* 755 490281 756 490107 757 125169 758 84057 759 133521 760 404775 761 913671* 762 48615 763 242445 764 141243 765 73059 766 988437* 767 315 768 65475 769 484455 770 354417 771 56199 772 743433 773 62391 774 173667 775 125385 776 26775 777 188979 778 410187 779 239271 780 51777 781 88299 782 406707 783 108351 784 364203 785 193515 786 58257 787 17481 788 20997 789 19485 790 116103 791 217809 792 488805 793 98649 794 18495 795 119259 796 212157 797 526701 798 679623 799 95565 800 207663 801 291951 802 353127 803 267795 804 442227 805 72861 806 613383 807 136119 808 142785 809 47055 810 539157 811 70539 812 191085 813 10125 814 105537 815 234315 816 385887 817 64401 818 789453 819 377451 820 125385 821 49041 822 640677 823 268101 824 66975 825 134481 826 515955 827 12285 828 78375 829 1365 830 554925 831 24609 832 524217 833 49539 834 130323 835 155085 836 1175493* 837 62361 838 127905 839 238395 840 916815 841 86679 842 129237 843 122685 844 87303 845 451209 846 61785 847 5265 848 255693 849 163965 850 278427 851 382875 852 53763 853 18885 854 169407 855 157251 856 87495 857 88095 858 63135 859 555039 860 629997 861 37359 862 798315 863 93765 864 722967 865 31575 866 1744257* 867 60681 868 483735 869 399591 870 167967 871 1767711* 872 111027 873 55209 874 75783 875 50565 876 272085 877 31005 878 296043 879 622671 880 4107 881 134511 882 17145 883 430389 884 25833 885 1097925 886 87465 887 895101 888 45675 889 59421 890 910923 891 149091 892 115845 893 248349 894 173283 895 133875 896 11925 897 498981 898 59925 899 68901 900 105177 901 109305 902 1039227 903 23901 904 141615 905 344949 906 24747 907 248781 908 88407 909 179091 910 107457 911 551979 912 313485 913 127689 914 5673 915 136881 916 106413 917 233349 918 163377 919 280929 920 367023 921 94629 922 10533 923 382035 924 773367 925 50595 926 143403 927 80139 928 79623 929 83271 930 424167 931 2035431* 932 116385 933 94335 934 41727 935 390099 936 52953 937 164829 938 165537 939 369381 940 5955 941 202335 942 112053 943 317955 944 164787 945 60729 946 170085 947 27825 948 61425 949 23805 950 13503 951 385695 952 178173 953 21741 954 45243 955 351765 956 232947 957 77805 958 66417 959 399105 960 770193 961 98061 962 312297 963 1170699 964 177255 965 7995 966 82995 967 703701 968 514437 969 77565 970 113745 971 1390269 972 493173 973 16011 974 192255 975 947859 976 68313 977 230439 978 582717 979 262575 980 441357 981 402141 982 626943 983 10485 984 163497 985 411081 986 706773 987 1305255 988 97323 989 349521 990 417375 991 234291 992 56685 993 179445 994 24963 995 219069 996 237675 997 400941 998 330075 999 586899 1000 467343 [/code][/quote] The value of n=1 should be k=2 instead of k=3, i.e.: 2*2^1-1=3 and 2*2^1+1=5. Trivial result...the only twin prime for an even k, as shown on my web page. Gary |
Gary/ Karsten
Decided to take a break from Very Prime Series and give a hand here. I have two machines (albeit with no CPU power) working on first instance (k) primes up to n=3000. Results of first instance primes to 1400 show a rather nice curve when plotting ln(k/n) against n. Best fit looks to be logarithmic as well. Would be interested to know if this might be a good way to target large twins. For example, [U]if[/U] the extrapolation of the best fit to n=333333, gave A= ln(k/n)= 9, then the first twin would be, on average, at k=2.70103*10^9 then the test might look at n from 333333 to n 333433, say at A=8.999 to 9.001 or k=2.69833*10^9 to 2.70373*10^9. A 50 million k range, sieved to 1T would provide about 4,000 candidates for prime checking or 400,000 overall for a 100 k range. Mind you I am not sure A= 9 is right for n=333333. But maybe someone could extrapolate and work out the odds of finding a twin in the suggested k/n matrix. Also the plot of ln(ln(k/n)) looks as if it heading to a value of 2 with over 87% of values between 1.5 and 2 for n from 1 to 1383, and 92% from n=1 to 1383. |
Oooer,
Think I am a long way out on my forecast for n=333333, but maybe mathematicians can come to the rescue!! |
Here are the results for 1000 to 2000, with asterisked jumping champions:
[CODE] 1001 282285 1002 1028307 1003 140691 1004 519915 1005 370605 1006 176877 1007 37275 1008 48225 1009 551679 1010 258483 1011 895221 1012 334137 1013 74091 1014 35523 1015 424125 1016 314217 1017 320931 1018 19887 1019 122595 1020 313023 1021 341871 1022 292215 1023 622485 1024 254697 1025 258621 1026 155925 1027 711885 1028 98493 1029 938931 1030 416883 1031 461889 1032 177 1033 364365 1034 16233 1035 130869 1036 328497 1037 71421 1038 145035 1039 249429 1040 356433 1041 200655 1042 255807 1043 373965 1044 1391775 1045 94065 1046 124743 1047 109809 1048 3885 1049 217365 1050 312147 1051 437955 1052 85845 1053 212265 1054 461175 1055 91755 1056 88515 1057 33405 1058 620595 1059 60099 1060 265593 1061 1063875 1062 126963 1063 122325 1064 243183 1065 290955 1066 90165 1067 85911 1068 193443 1069 132525 1070 70623 1071 2092731* 1072 142443 1073 291231 1074 371997 1075 13131 1076 50025 1077 181521 1078 419145 1079 143781 1080 193347 1081 427611 1082 533187 1083 818961 1084 15315 1085 156615 1086 69417 1087 161289 1088 117225 1089 121719 1090 393687 1091 773475 1092 147417 1093 651939 1094 188793 1095 803589 1096 701433 1097 116805 1098 58287 1099 342189 1100 355065 1101 384981 1102 27435 1103 170115 1104 4275 1105 738375 1106 300135 1107 48681 1108 413103 1109 3525165* 1110 11007 1111 2439999 1112 111765 1113 196455 1114 843003 1115 1041429 1116 175803 1117 819225 1118 528783 1119 155445 1120 121365 1121 465891 1122 60513 1123 831105 1124 1504797 1125 148935 1126 1098363 1127 152835 1128 224973 1129 508131 1130 580923 1131 518091 1132 850785 1133 142125 1134 83205 1135 618735 1136 317793 1137 221979 1138 798135 1139 168195 1140 282873 1141 269745 1142 11007 1143 1057989 1144 162177 1145 1037355 1146 184923 1147 251175 1148 1398117 1149 487785 1150 232755 1151 1617651 1152 336315 1153 827451 1154 729543 1155 256491 1156 1035 1157 312651 1158 1120767 1159 1108365 1160 167013 1161 723411 1162 341427 1163 811269 1164 228093 1165 798891 1166 258117 1167 54339 1168 62823 1169 353679 1170 1053327 1171 300585 1172 1524225 1173 52461 1174 1078677 1175 46791 1176 147063 1177 964179 1178 337287 1179 613029 1180 830835 1181 261855 1182 656877 1183 924855 1184 396843 1185 1647891 1186 320223 1187 1173099 1188 65613 1189 955089 1190 154605 1191 1066371 1192 625365 1193 84435 1194 176085 1195 175785 1196 324915 1197 45201 1198 185157 1199 268191 1200 724545 1201 544929 1202 1055583 1203 2016501 1204 121275 1205 356415 1206 273957 1207 869541 1208 990165 1209 537369 1210 463983 1211 994269 1212 666495 1213 1280439 1214 2475495 1215 364455 1216 130803 1217 14199 1218 1267887 1219 188589 1220 825105 1221 88329 1222 1023483 1223 542151 1224 775365 1225 652335 1226 1653435 1227 662799 1228 79203 1229 42399 1230 3452103 1231 753849 1232 988593 1233 389895 1234 513813 1235 1115085 1236 1392603 1237 74109 1238 252753 1239 3267021 1240 334293 1241 13629 1242 298797 1243 529719 1244 64233 1245 88575 1246 621147 1247 3058299 1248 129315 1249 243819 1250 597465 1251 75705 1252 206655 1253 515679 1254 2396175 1255 763701 1256 53865 1257 187521 1258 635025 1259 484341 1260 147387 1261 32265 1262 2045985 1263 979605 1264 467595 1265 533805 1266 774807 1267 95229 1268 817797 1269 802431 1270 71805 1271 682881 1272 50655 1273 235845 1274 95847 1275 456549 1276 475515 1277 613971 1278 388005 1279 152091 1280 1318815 1281 2160741 1282 99105 1283 3124995 1284 426657 1285 322041 1286 9183 1287 5629461* 1288 1793223 1289 808101 1290 230085 1291 114105 1292 1462605 1293 186921 1294 341367 1295 49119 1296 1186755 1297 764001 1298 248163 1299 3339 1300 264915 1301 275619 1302 734325 1303 633381 1304 205503 1305 190881 1306 499917 1307 163731 1308 1501665 1309 270075 1310 384993 1311 1033569 1312 15657 1313 571581 1314 46965 1315 471795 1316 580623 1317 1627791 1318 438075 1319 304065 1320 1733043 1321 1065 1322 230223 1323 1033569 1324 969387 1325 62265 1326 996987 1327 2625 1328 1553745 1329 1281075 1330 754215 1331 3307209 1332 209853 1333 132225 1334 1143255 1335 480165 1336 1199463 1337 560391 1338 92115 1339 142185 1340 1857213 1341 422385 1342 1224117 1343 117975 1344 1280295 1345 351015 1346 853875 1347 582729 1348 1252053 1349 919881 1350 245325 1351 878415 1352 1236897 1353 263571 1354 38565 1355 62289 1356 998823 1357 158355 1358 1416285 1359 313551 1360 456645 1361 268485 1362 158043 1363 998205 1364 471537 1365 365691 1366 769293 1367 7755 1368 682743 1369 249285 1370 290133 1371 518529 1372 395787 1373 188475 1374 1108203 1375 125601 1376 147777 1377 1157415 1378 2088045 1379 570699 1380 1462017 1381 856821 1382 941577 1383 67821 1384 751497 1385 106449 1386 708885 1387 175905 1388 107205 1389 97899 1390 14877 1391 76479 1392 1984383 1393 370749 1394 88155 1395 870741 1396 1342773 1397 542805 1398 658935 1399 1186401 1400 797373 1401 878385 1402 80103 1403 1414851 1404 761427 1405 891891 1406 18003 1407 2122659 1408 98475 1409 174009 1410 939915 1411 173229 1412 2561475 1413 1041609 1414 893937 1415 207519 1416 726975 1417 2445081 1418 285153 1419 689589 1420 1655163 1421 203949 1422 797955 1423 1522551 1424 136065 1425 86205 1426 232365 1427 1946031 1428 2266035 1429 170379 1430 438645 1431 75519 1432 379317 1433 297591 1434 1132857 1435 2376429 1436 961797 1437 1592955 1438 686343 1439 1993755 1440 10083 1441 74031 1442 702927 1443 613095 1444 316275 1445 1140285 1446 93837 1447 51651 1448 518787 1449 575949 1450 650757 1451 769209 1452 1122567 1453 632541 1454 442635 1455 849261 1456 3351873 1457 515535 1458 611433 1459 113931 1460 1605915 1461 141099 1462 15927 1463 1233519 1464 455445 1465 114681 1466 74505 1467 1252569 1468 85077 1469 571071 1470 273765 1471 81489 1472 131025 1473 330201 1474 1524045 1475 66381 1476 751005 1477 129645 1478 1114995 1479 385971 1480 677613 1481 3420999 1482 1436103 1483 57891 1484 167745 1485 377349 1486 704373 1487 1912131 1488 524727 1489 1231161 1490 3160707 1491 338649 1492 1267167 1493 1246179 1494 471945 1495 187509 1496 521073 1497 1408449 1498 2480883 1499 306939 1500 32547 1501 1778709 1502 652323 1503 2676891 1504 1424085 1505 1327329 1506 910305 1507 323895 1508 132093 1509 86361 1510 2608005 1511 1934409 1512 858315 1513 577179 1514 1200675 1515 3154659 1516 303345 1517 2700279 1518 1270677 1519 937071 1520 3027615 1521 381699 1522 5690025* 1523 133725 1524 187155 1525 1759851 1526 302415 1527 362619 1528 465015 1529 1916355 1530 137427 1531 519765 1532 83973 1533 36045 1534 2798223 1535 263505 1536 1682853 1537 587505 1538 1581567 1539 568809 1540 369837 1541 1562889 1542 318075 1543 442191 1544 606537 1545 135831 1546 657417 1547 286581 1548 583275 1549 3521211 1550 427143 1551 447711 1552 297627 1553 291 1554 1979757 1555 1585425 1556 81255 1557 349395 1558 924657 1559 2046261 1560 108885 1561 1726149 1562 201645 1563 2062539 1564 1439853 1565 882459 1566 42507 1567 601515 1568 1326975 1569 1108371 1570 935943 1571 822555 1572 715995 1573 324129 1574 424617 1575 620055 1576 246045 1577 586755 1578 118893 1579 238509 1580 4894863 1581 763935 1582 285423 1583 682179 1584 849357 1585 339621 1586 566475 1587 1013535 1588 142197 1589 1483749 1590 118773 1591 1408785 1592 1439703 1593 239745 1594 360513 1595 303939 1596 1277775 1597 1704015 1598 937233 1599 15375 1600 2037615 1601 128619 1602 2329245 1603 89895 1604 4199007 1605 1274121 1606 923853 1607 3542361 1608 992037 1609 390621 1610 222783 1611 2597511 1612 3516297 1613 417711 1614 219693 1615 876975 1616 78327 1617 799335 1618 605283 1619 133155 1620 2564265 1621 942615 1622 1283757 1623 33549 1624 832413 1625 65835 1626 501447 1627 1149735 1628 1153323 1629 2937105 1630 166173 1631 2852121 1632 1530903 1633 202551 1634 216573 1635 1623171 1636 264327 1637 4553769 1638 213897 1639 517215 1640 34215 1641 164271 1642 620433 1643 3030975 1644 1387347 1645 2923179 1646 1206135 1647 183855 1648 160587 1649 269541 1650 1873503 1651 884751 1652 33957 1653 323031 1654 682113 1655 876849 1656 1595433 1657 566445 1658 3824937 1659 1813611 1660 20733 1661 589281 1662 260757 1663 1216815 1664 1183377 1665 885909 1666 3120063 1667 2131941 1668 1262193 1669 1022901 1670 960075 1671 552771 1672 56685 1673 1112175 1674 1859853 1675 917541 1676 96897 1677 24969 1678 34725 1679 520719 1680 832947 1681 602331 1682 292407 1683 1367541 1684 3336843 1685 239715 1686 2610675 1687 228111 1688 153513 1689 27765 1690 674133 1691 598875 1692 1520523 1693 747711 1694 401055 1695 570831 1696 592563 1697 1189089 1698 1627455 1699 285861 1700 241455 1701 2189229 1702 5434023 1703 1377975 1704 151827 1705 312405 1706 1362567 1707 1403001 1708 270633 1709 497991 1710 3470667 1711 1458675 1712 635163 1713 213201 1714 1036635 1715 4239039 1716 5862777* 1717 1717869 1718 51747 1719 386061 1720 3976863 1721 45951 1722 1887375 1723 650859 1724 379323 1725 787395 1726 435345 1727 1444089 1728 1652577 1729 174915 1730 3604917 1731 970335 1732 1796067 1733 840315 1734 2362815 1735 1639251 1736 679977 1737 620571 1738 2231685 1739 595581 1740 377787 1741 872055 1742 1824297 1743 967425 1744 1424145 1745 260289 1746 2503953 1747 2776215 1748 645327 1749 2970279 1750 274323 1751 373185 1752 1260717 1753 3720729 1754 1073577 1755 2880981 1756 2120643 1757 41229 1758 1585017 1759 300351 1760 3314793 1761 289095 1762 402687 1763 29481 1764 984987 1765 426105 1766 447465 1767 84159 1768 1820583 1769 285009 1770 1596045 1771 691419 1772 333663 1773 2318871 1774 3654177 1775 1090125 1776 595053 1777 358875 1778 602337 1779 842529 1780 2161035 1781 2065071 1782 219495 1783 851709 1784 2669145 1785 243399 1786 42825 1787 1118181 1788 2421057 1789 428631 1790 383313 1791 113025 1792 1904805 1793 95151 1794 98583 1795 453741 1796 2079255 1797 2800941 1798 1172337 1799 2568609 1800 732567 1801 4073949 1802 3368655 1803 2079045 1804 1003533 1805 674505 1806 1206513 1807 474249 1808 211527 1809 2702121 1810 381687 1811 1009869 1812 994107 1813 948789 1814 153537 1815 2490609 1816 119217 1817 1262079 1818 2999217 1819 5680095 1820 79335 1821 208101 1822 2517873 1823 57495 1824 1047153 1825 2028459 1826 943863 1827 943941 1828 5392107 1829 3421551 1830 2098275 1831 1688079 1832 1186437 1833 4735311 1834 529983 1835 535395 1836 1230183 1837 1388661 1838 434085 1839 584829 1840 2480025 1841 525759 1842 560313 1843 734361 1844 2615235 1845 4198689 1846 1177785 1847 1019379 1848 674847 1849 87585 1850 4155033 1851 317649 1852 626703 1853 4020069 1854 699963 1855 203211 1856 2185443 1857 3799731 1858 29835 1859 1685445 1860 13317 1861 331731 1862 843273 1863 4007301 1864 166353 1865 508911 1866 2739747 1867 475719 1868 1142103 1869 82275 1870 311067 1871 1838235 1872 845973 1873 569451 1874 226293 1875 1002141 1876 3964035 1877 2332491 1878 1327095 1879 1345965 1880 22035 1881 354189 1882 1104123 1883 919335 1884 563217 1885 191505 1886 531717 1887 2096265 1888 800457 1889 3173061 1890 3049323 1891 922749 1892 2294307 1893 849849 1894 249417 1895 417045 1896 4020993 1897 413271 1898 675255 1899 1567515 1900 4425 1901 441531 1902 1436775 1903 435609 1904 3037815 1905 520215 1906 328557 1907 589455 1908 1018797 1909 459291 1910 1456293 1911 4789059 1912 4947147 1913 426615 1914 2018727 1915 2202885 1916 558183 1917 1542735 1918 1434207 1919 164835 1920 588453 1921 1274835 1922 362613 1923 228525 1924 935265 1925 1019949 1926 891273 1927 826575 1928 1216137 1929 1132989 1930 426165 1931 1126005 1932 298965 1933 39171 1934 605973 1935 2497911 1936 101505 1937 485181 1938 2986623 1939 245481 1940 1342107 1941 2349669 1942 2730237 1943 819189 1944 335097 1945 704409 1946 4290147 1947 5837991 1948 1731387 1949 1217601 1950 1105407 1951 928965 1952 491337 1953 197139 1954 8007 1955 396669 1956 616893 1957 700569 1958 1550433 1959 1748631 1960 1373625 1961 2612919 1962 303147 1963 822435 1964 1738983 1965 3094341 1966 63237 1967 1843419 1968 1330095 1969 637341 1970 570597 1971 3885 1972 295503 1973 2244459 1974 3217737 1975 1017381 1976 2980953 1977 4103841 1978 1088085 1979 3564789 1980 3550827 1981 800109 1982 2046087 1983 182319 1984 429777 1985 31545 1986 880317 1987 3175779 1988 207213 1989 103239 1990 2965107 1991 2852079 1992 1963113 1993 1301685 1994 1780785 1995 1314885 1996 284655 1997 2378895 1998 377085 1999 3400089 2000 4605615 [/CODE] |
hey, great work robert!
i will include them here [url]www.rieselprime.org/FirstKTwin.htm[/url] when i got time! thanks! |
Here are first instance twins from n=2000 to n=3000. Now a good way towards n=4000
[code] n first k 2001 133641 2002 649683 2003 2930049 2004 1427463 2005 411801 2006 595665 2007 1154619 2008 1100385 2009 934875 2010 667275 2011 954759 2012 606633 2013 2595345 2014 166845 2015 1190745 2016 1527327 2017 4675869 2018 720387 2019 2345859 2020 3118845 2021 2558685 2022 622365 2023 1517451 2024 10095 2025 3632955 2026 773193 2027 2129109 2028 2235525 2029 1885245 2030 111483 2031 1974135 2032 5717397 2033 1332465 2034 852807 2035 1580595 2036 1151973 2037 451959 2038 1018803 2039 181635 2040 615927 2041 3436125 2042 390345 2043 1551489 2044 395835 2045 2897031 2046 689187 2047 3666651 2048 442533 2049 2094795 2050 2451435 2051 243309 2052 260715 2053 1735629 2054 3096345 2055 1001985 2056 1874007 2057 784581 2058 4510305 2059 2444265 2060 329613 2061 357159 2062 7147317* 2063 1119579 2064 2763243 2065 825315 2066 4315755 2067 1284945 2068 1066533 2069 2132229 2070 1902303 2071 176829 2072 2144103 2073 606879 2074 687363 2075 1871289 2076 457005 2077 2539431 2078 1590315 2079 1473615 2080 1370817 2081 2558949 2082 2970705 2083 84609 2084 1662657 2085 2956539 2086 542673 2087 2364531 2088 2206503 2089 1019235 2090 3338835 2091 3359289 2092 1375155 2093 281415 2094 3084393 2095 2336349 2096 189837 2097 123165 2098 1733655 2099 3562065 2100 191823 2101 161511 2102 757173 2103 2760621 2104 1676493 2105 1231005 2106 423327 2107 1070481 2108 8173875* 2109 981171 2110 251235 2111 3498669 2112 59553 2113 661341 2114 3489753 2115 1983459 2116 269877 2117 1766541 2118 374757 2119 2165961 2120 969993 2121 368379 2122 966537 2123 3632469 2124 2199687 2125 927249 2126 1339323 2127 1189311 2128 373077 2129 11655 2130 1796025 2131 1320969 2132 387015 2133 3241005 2134 4231827 2135 403971 2136 794463 2137 358101 2138 40215 2139 808635 2140 1206507 2141 3789009 2142 2047887 2143 723555 2144 215307 2145 557925 2146 2486253 2147 623661 2148 647787 2149 878571 2150 959757 2151 4665759 2152 250977 2153 2285889 2154 4471203 2155 401025 2156 769737 2157 380499 2158 4913103 2159 477075 2160 5649675 2161 1045821 2162 33117 2163 2489541 2164 1547013 2165 1314579 2166 1123593 2167 6311811 2168 2865393 2169 408471 2170 1281513 2171 2061111 2172 467577 2173 2655471 2174 1231593 2175 3716169 2176 1172205 2177 3578565 2178 287103 2179 8715615* 2180 480777 2181 706431 2182 53955 2183 7423941 2184 1803315 2185 66729 2186 810897 2187 1295271 2188 2439885 2189 602595 2190 4485507 2191 4359 2192 1844283 2193 425031 2194 4654305 2195 179745 2196 24405 2197 4499175 2198 895257 2199 6930819 2200 4012377 2201 2984799 2202 903375 2203 1317555 2204 640707 2205 806271 2206 2327703 2207 603495 2208 2644593 2209 1214331 2210 1505997 2211 436515 2212 1612533 2213 6201 2214 1405647 2215 3203205 2216 3028725 2217 879501 2218 863343 2219 416721 2220 13939725* 2221 1566255 2222 3628233 2223 1892385 2224 3047007 2225 976011 2226 923205 2227 6554205 2228 2935587 2229 2972529 2230 2126025 2231 5493585 2232 3593625 2233 2826081 2234 509823 2235 3665961 2236 113685 2237 644385 2238 143427 2239 1264095 2240 2389677 2241 1436769 2242 2657433 2243 823179 2244 3927957 2245 963429 2246 141165 2247 908919 2248 1814997 2249 1636011 2250 1208235 2251 1513965 2252 3355227 2253 75219 2254 7286685 2255 7419 2256 488733 2257 109599 2258 402567 2259 2707365 2260 5754897 2261 538905 2262 2540235 2263 1268265 2264 334863 2265 1755045 2266 619695 2267 812025 2268 1441893 2269 4040391 2270 684123 2271 719925 2272 1486575 2273 378291 2274 1294263 2275 211365 2276 5279163 2277 255999 2278 43947 2279 2458665 2280 25293 2281 3015531 2282 1120767 2283 1025139 2284 11455353 2285 3492885 2286 3932037 2287 3226839 2288 3184605 2289 928329 2290 3743943 2291 1571535 2292 6518367 2293 117705 2294 4774575 2295 2711271 2296 2494323 2297 1601151 2298 965373 2299 1155231 2300 4387947 2301 261609 2302 3444303 2303 545955 2304 3792927 2305 533079 2306 4193055 2307 1764471 2308 848127 2309 2331879 2310 1491345 2311 257319 2312 723477 2313 5870889 2314 470625 2315 2635581 2316 1501227 2317 298599 2318 149955 2319 1017279 2320 1449825 2321 625821 2322 3600657 2323 2401035 2324 825027 2325 2895951 2326 1821477 2327 1671075 2328 143925 2329 1786041 2330 900645 2331 3034155 2332 3049125 2333 43089 2334 2249907 2335 554811 2336 7262457 2337 2613825 2338 1122063 2339 1208025 2340 131373 2341 3225699 2342 1730217 2343 4251891 2344 1293297 2345 4404261 2346 2074233 2347 3894099 2348 1607295 2349 256359 2350 4011873 2351 3974571 2352 2129943 2353 756471 2354 1179975 2355 5739951 2356 3090843 2357 3325329 2358 660993 2359 2272311 2360 2209227 2361 140319 2362 808935 2363 900021 2364 486087 2365 606621 2366 3500433 2367 988899 2368 1419837 2369 410799 2370 231915 2371 1438809 2372 2782503 2373 2746149 2374 420267 2375 403521 2376 194367 2377 2408769 2378 5432883 2379 317145 2380 459807 2381 1413381 2382 1804275 2383 1068699 2384 1526997 2385 2721639 2386 1397865 2387 604989 2388 2572503 2389 699531 2390 1073703 2391 820761 2392 936465 2393 5188569 2394 554055 2395 6620265 2396 1045113 2397 451845 2398 540057 2399 3925809 2400 12123027 2401 753489 2402 2858337 2403 9618801 2404 2873163 2405 687099 2406 6860187 2407 859329 2408 162183 2409 942441 2410 913425 2411 2101551 2412 2205837 2413 591579 2414 684813 2415 650709 2416 708897 2417 4326885 2418 2249355 2419 6996435 2420 399027 2421 1501815 2422 5330073 2423 617295 2424 351237 2425 2331489 2426 1700745 2427 2249355 2428 1426815 2429 379401 2430 3521445 2431 2713935 2432 10912485 2433 3965685 2434 11808915 2435 740181 2436 4112013 2437 1503909 2438 11136945 2439 1511241 2440 1282917 2441 379005 2442 3267045 2443 1231011 2444 1565493 2445 2883729 2446 6014253 2447 1715181 2448 5903835 2449 6606249 2450 1273023 2451 10357665 2452 175245 2453 6871845 2454 3140505 2455 11621685 2456 484047 2457 708795 2458 778317 2459 1261359 2460 331545 2461 2476005 2462 2178705 2463 1972995 2464 4438875 2465 2459565 2466 7927227 2467 1052541 2468 2009427 2469 605901 2470 60957 2471 1469061 2472 1557105 2473 52935 2474 271947 2475 3044661 2476 572223 2477 4678059 2478 2677083 2479 845949 2480 2485287 2481 639159 2482 1859067 2483 4125519 2484 1958313 2485 775749 2486 568767 2487 150111 2488 1874823 2489 804801 2490 4571175 2491 1925415 2492 1934517 2493 1275999 2494 1869555 2495 398385 2496 7484367 2497 766611 2498 56727 2499 1694595 2500 575745 2501 86085 2502 416007 2503 3248349 2504 1706667 2505 562809 2506 244407 2507 1906821 2508 2473527 2509 3935829 2510 1539285 2511 2696031 2512 705795 2513 1694475 2514 3419607 2515 812355 2516 343395 2517 6644109 2518 94815 2519 1857729 2520 4990947 2521 1296699 2522 245895 2523 878481 2524 4332453 2525 2165169 2526 1427127 2527 3986631 2528 6039603 2529 33939 2530 1311975 2531 9119961 2532 2946213 2533 1834041 2534 1990185 2535 207939 2536 2664915 2537 3206295 2538 731673 2539 2443209 2540 2645943 2541 454989 2542 1231077 2543 3092235 2544 5280615 2545 1466271 2546 3537933 2547 4499871 2548 258417 2549 1182471 2550 4024653 2551 1694625 2552 4992807 2553 1204719 2554 106455 2555 4458441 2556 2796033 2557 1669449 2558 2196375 2559 4229589 2560 100995 2561 2443431 2562 260295 2563 903861 2564 3509157 2565 621045 2566 1949103 2567 114201 2568 1097505 2569 79029 2570 780807 2571 886329 2572 712215 2573 4769349 2574 475863 2575 704649 2576 416937 2577 2264481 2578 1247493 2579 1081539 2580 590517 2581 2458455 2582 181287 2583 5741505 2584 3556737 2585 2617089 2586 348873 2587 4053015 2588 1299693 2589 8371671 2590 1045233 2591 372849 2592 2753643 2593 1696365 2594 1162773 2595 671781 2596 6094713 2597 183039 2598 3030753 2599 236049 2600 5675475 2601 2153811 2602 190815 2603 3151935 2604 1817643 2605 1666821 2606 12924117 2607 4004469 2608 2879193 2609 139989 2610 426045 2611 662241 2612 934683 2613 1333929 2614 5152947 2615 2439525 2616 4649697 2617 3453555 2618 5695893 2619 2897379 2620 193845 2621 4290441 2622 1154877 2623 5846175 2624 2032425 2625 1030431 2626 736773 2627 377685 2628 577365 2629 1129701 2630 1323357 2631 3091221 2632 589335 2633 410001 2634 166605 2635 1122171 2636 595365 2637 89115 2638 5478003 2639 3811335 2640 985215 2641 2310279 2642 2835045 2643 4563471 2644 573627 2645 4521171 2646 1888575 2647 1511181 2648 2547693 2649 4619499 2650 3437817 2651 1015815 2652 1216857 2653 1221885 2654 7059063 2655 2516745 2656 2727627 2657 5563359 2658 3372297 2659 1835409 2660 240495 2661 3673761 2662 1500213 2663 1548429 2664 3605595 2665 1573299 2666 968583 2667 7326225 2668 1080345 2669 1259829 2670 2277663 2671 2420439 2672 3083613 2673 11958531 2674 4792023 2675 1161735 2676 4933443 2677 643641 2678 10203777 2679 61269 2680 1436325 2681 2024211 2682 8889465 2683 4113411 2684 4908435 2685 93429 2686 3934047 2687 3287829 2688 7006263 2689 1060011 2690 1041795 2691 398589 2692 7944825 2693 964149 2694 1350795 2695 59415 2696 902343 2697 1074075 2698 6025185 2699 1113525 2700 1925445 2701 1549605 2702 6286287 2703 5440785 2704 2354745 2705 1053399 2706 977427 2707 32811 2708 5547693 2709 5487639 2710 481605 2711 437835 2712 1341195 2713 1060089 2714 1808097 2715 1783239 2716 1843857 2717 16812351* 2718 5734713 2719 2510079 2720 482103 2721 6000501 2722 1479627 2723 2945835 2724 2860125 2725 1531395 2726 792537 2727 3262599 2728 3397107 2729 3935889 2730 2321433 2731 7364859 2732 14301783 2733 3360051 2734 1315497 2735 2263065 2736 14572653 2737 6372735 2738 481545 2739 6473169 2740 340065 2741 2700729 2742 421845 2743 52011 2744 1766127 2745 1049505 2746 1573557 2747 1362159 2748 84255 2749 9000045 2750 2069775 2751 3219225 2752 2133207 2753 2330175 2754 1615203 2755 213255 2756 2106765 2757 11027769 2758 2996775 2759 238881 2760 1667967 2761 5267661 2762 3044367 2763 5110275 2764 7759845 2765 2507301 2766 1068525 2767 292209 2768 11093037 2769 566409 2770 661635 2771 3098649 2772 640455 2773 5088489 2774 2575803 2775 467661 2776 617433 2777 4566309 2778 2330307 2779 556779 2780 14337705 2781 3220305 2782 689613 2783 2996889 2784 5921757 2785 5956461 2786 9335997 2787 4640235 2788 3797235 2789 1327221 2790 4215237 2791 4254195 2792 2704953 2793 630081 2794 1293315 2795 17255511* 2796 727563 2797 139755 2798 5603163 2799 5940429 2800 1967433 2801 2207439 2802 458055 2803 1702281 2804 2313933 2805 288015 2806 3456585 2807 16158471 2808 466455 2809 2227269 2810 1106493 2811 5634765 2812 1217523 2813 3071601 2814 5801007 2815 2203269 2816 2200113 2817 3194055 2818 204357 2819 1695369 2820 2646273 2821 6075 2822 287217 2823 288615 2824 4327503 2825 923289 2826 2103813 2827 20805 2828 909843 2829 1550049 2830 2468193 2831 2106699 2832 1014393 2833 1653951 2834 61947 2835 2778309 2836 6207087 2837 3304581 2838 10169775 2839 884205 2840 1612197 2841 4633191 2842 3495975 2843 4715439 2844 58053 2845 627381 2846 10725 2847 511065 2848 1123515 2849 935529 2850 1309137 2851 323601 2852 815517 2853 3369525 2854 2537325 2855 4427595 2856 4030533 2857 1534119 2858 3585705 2859 1536981 2860 4051923 2861 368595 2862 614187 2863 1922361 2864 3812463 2865 3370299 2866 2223795 2867 36159 2868 705933 2869 1398321 2870 1571097 2871 6877731 2872 512253 2873 6672081 2874 3596667 2875 539421 2876 800757 2877 8503521 2878 1886403 2879 13157931 2880 7072905 2881 3142449 2882 684015 2883 627735 2884 1975953 2885 5101209 2886 2311773 2887 88629 2888 275325 2889 2333019 2890 6826905 2891 3037191 2892 6122265 2893 4464465 2894 980265 2895 2823849 2896 1508925 2897 4519521 2898 1095363 2899 69735 2900 1371135 2901 5099181 2902 157893 2903 696405 2904 9783483 2905 328545 2906 11411613 2907 1678275 2908 2838123 2909 1841685 2910 4322745 2911 3445101 2912 791253 2913 2804205 2914 14275353 2915 6891045 2916 1539597 2917 11005071 2918 12413187 2919 6586779 2920 1629573 2921 1202271 2922 4129863 2923 5256165 2924 1206873 2925 836325 2926 2107245 2927 1020141 2928 1763907 2929 6573231 2930 4244325 2931 757491 2932 2291283 2933 7475019 2934 3297507 2935 8714775 2936 3430953 2937 2713569 2938 1426413 2939 6322515 2940 1538163 2941 2765475 2942 6645657 2943 7253025 2944 2681937 2945 12195 2946 1294725 2947 809979 2948 3103845 2949 4511325 2950 7214457 2951 4022535 2952 5410377 2953 1721469 2954 3229527 2955 3246201 2956 265977 2957 3353631 2958 2484813 2959 4155909 2960 7358085 2961 1303029 2962 4964043 2963 1839639 2964 3869667 2965 1957191 2966 429717 2967 3586251 2968 1604385 2969 3225009 2970 1524735 2971 8492799 2972 1057923 2973 482991 2974 6391455 2975 584529 2976 2337513 2977 1918329 2978 385815 2979 1990299 2980 2580915 2981 561669 2982 149013 2983 1473291 2984 547623 2985 962211 2986 5351445 2987 4388559 2988 4109163 2989 2880939 2990 6570627 2991 4028841 2992 4792815 2993 224025 2994 5546193 2995 1838061 2996 3545457 2997 1325205 2998 4013547 2999 2016735 3000 1298973 [/CODE] |
all data are now available here: [url]www.rieselprime.org/FirstKTwin.htm[/url].
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