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Difference between revisions of "Williams prime PP least"
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| − | {{DISPLAYTITLE:Williams primes of the form {{Kbn|+|(b+1)|b|n}}, least | + | {{DISPLAYTITLE:Williams primes of the form {{Kbn|+|(b+1)|b|n}}, least {{Vn}}-values}} |
==Description== | ==Description== | ||
| − | Here are shown the least | + | Here are shown the least {{Vn}}-value ≥ 1 for any base {{Vb}} with 2 ≤ {{Vb}} ≤ 1024 which generates a [[Williams prime]] of the form {{Kbn|+|(b+1)|b|n}}. |
| + | |||
| + | Values of bases {{Vb}} ≡ 1 mod 3 are always divisible by 3, so not listed here. | ||
==History== | ==History== | ||
| − | * | + | *{{HistF|2020-07-19|135981 ([[Williams prime PP 327|b=327]])|(unknown),Conjectures 'R Us|551009}} |
| + | *{{HistF|2019-06-15|3615 ([[Williams prime PP 929|b=929]])|Karsten Bonath}} | ||
| + | *{{HistF|2019-06-15|4917 ([[Williams prime PP 912|b=912]])|Karsten Bonath}} | ||
| + | *{{HistF|2019-05-21|13418 ([[Williams prime PP 719|b=719]])|Karsten Bonath}} | ||
| + | *{{HistF|2019-05-21|37506 ([[Williams prime PP 717|b=717]])|Karsten Bonath}} | ||
| + | *{{HistF|2019-05-20|24110 ([[Williams prime PP 710|b=710]])|Karsten Bonath}} | ||
| + | *{{HistF|2019-05-15|3521 ([[Williams prime PP 473|b=473]])|Karsten Bonath}} | ||
| + | *{{HistF|2019-05-15|1806 ([[Williams prime PP 542|b=542]])|Karsten Bonath}} | ||
| + | *{{HistF|2019-05-15|6774 ([[Williams prime PP 683|b=683]])|Karsten Bonath}} | ||
| + | *{{HistF|2019-05-14|11229 ([[Williams prime PP 425|b=425]])|Karsten Bonath}} | ||
| − | ==Wanted values== | + | ==Wanted values [searched range]== |
| − | + | <div style="column-count:6;-moz-column-count:6;-webkit-column-count:6"> | |
| − | : | + | {{#dpl: |
| + | |debug=1 | ||
| + | |namespace= | ||
| + | |category=Williams prime PP without | ||
| + | |include={Williams prime}:WiBase,{Williams prime}:WiBase,{Williams prime}:WiMaxn | ||
| + | |mode=userformat | ||
| + | |format=<ul type="disc">,<li>,</li>,</ul> | ||
| + | |secseparators=[[Williams prime PP ,|,,]], [,] | ||
| + | |order=ascending | ||
| + | |ordermethod=sortkey | ||
| + | |oneresultheader=<b>There is one sequence</b>: | ||
| + | |resultsheader=<b>There are %PAGES% sequences</b>: | ||
| + | |noresultsheader=<b>There are no sequences.</b> | ||
| + | }}</div> | ||
| + | '''Notes:''' | ||
| + | *Base 1017 is sometimes reserved and worked on by [[Conjectures 'R Us]]. | ||
==Data== | ==Data== | ||
| Line 15: | Line 41: | ||
<div style="width: 12em; height:50em; overflow:auto;"> | <div style="width: 12em; height:50em; overflow:auto;"> | ||
{| class="wikitable sortable" style="height: 200px" | {| class="wikitable sortable" style="height: 200px" | ||
| − | ! Base | + | ! data-sort-type="number" class="fixhead" | Base |
| − | ! | + | ! data-sort-type="number" class="fixhead" | {{Vn}}-value {{#for_external_table:<nowiki/> |
{{!}}- | {{!}}- | ||
{{!}} style="text-align:right" {{!}} {{{b}}} | {{!}} style="text-align:right" {{!}} {{{b}}} | ||
| Line 23: | Line 49: | ||
|} | |} | ||
</div> | </div> | ||
| − | + | {{Navbox Williams primes}} | |
| − | |||
[[Category:Williams prime PP| ]] | [[Category:Williams prime PP| ]] | ||
Latest revision as of 00:47, 8 September 2024
Description
Here are shown the least n-value ≥ 1 for any base b with 2 ≤ b ≤ 1024 which generates a Williams prime of the form (b+1)•bn+1.
Values of bases b ≡ 1 mod 3 are always divisible by 3, so not listed here.
History
- 2020-07-19: Found n = 135981 (b=327), (unknown), Conjectures 'R Us
- 2019-06-15: Found n = 3615 (b=929), Karsten Bonath
- 2019-06-15: Found n = 4917 (b=912), Karsten Bonath
- 2019-05-21: Found n = 13418 (b=719), Karsten Bonath
- 2019-05-21: Found n = 37506 (b=717), Karsten Bonath
- 2019-05-20: Found n = 24110 (b=710), Karsten Bonath
- 2019-05-15: Found n = 3521 (b=473), Karsten Bonath
- 2019-05-15: Found n = 1806 (b=542), Karsten Bonath
- 2019-05-15: Found n = 6774 (b=683), Karsten Bonath
- 2019-05-14: Found n = 11229 (b=425), Karsten Bonath
Wanted values [searched range]
Notes:
- Base 1017 is sometimes reserved and worked on by Conjectures 'R Us.
Data
The data file can be found here.
| Base | n-value |
|---|---|
| 2 | 1 |
| 3 | 1 |
| 5 | 1 |
| 6 | 1 |
| 8 | 1 |
| 9 | 2 |
| 11 | 2 |
| 12 | 1 |
| 14 | 1 |
| 15 | 1 |
| 17 | 1 |
| 18 | 9 |
| 20 | 1 |
| 21 | 1 |
| 23 | 2 |
| 24 | 1 |
| 26 | 2 |
| 27 | 1 |
| 29 | 5 |
| 30 | 2 |
| 32 | 5 |
| 33 | 1 |
| 35 | 2 |
| 36 | 3 |
| 38 | 1 |
| 39 | 3 |
| 41 | 1 |
| 42 | 2 |
| 44 | 2 |
| 45 | 2 |
| 47 | 2 |
| 48 | 6 |
| 50 | 1 |
| 51 | 183 |
| 53 | 2 |
| 54 | 1 |
| 56 | 2 |
| 57 | 1 |
| 59 | 1 |
| 60 | 21 |
| 62 | 1 |
| 63 | 185 |
| 65 | 3 |
| 66 | 1 |
| 68 | 2 |
| 69 | 1 |
| 71 | 1 |
| 72 | 120 |
| 74 | 2 |
| 75 | 1 |
| 77 | 1 |
| 78 | 1 |
| 80 | 1 |
| 81 | 8 |
| 83 | 5 |
| 84 | 9 |
| 86 | 2 |
| 87 | 2 |
| 89 | 1 |
| 90 | 1 |
| 92 | 2 |
| 93 | 3 |
| 95 | 9 |
| 96 | 14 |
| 98 | 3 |
| 99 | 1 |
| 101 | 1 |
| 102 | 5 |
| 104 | 5 |
| 105 | 1 |
| 107 | 80 |
| 108 | 398 |
| 110 | 1 |
| 111 | 1 |
| 113 | 3 |
| 114 | 3 |
| 116 | 44 |
| 117 | 1 |
| 119 | 1 |
| 120 | 2 |
| 122 | 2 |
| 123 | 2 |
| 125 | 2 |
| 126 | 5 |
| 128 | 3 |
| 129 | 54 |
| 131 | 1 |
| 132 | 54 |
| 134 | 9 |
| 135 | 17 |
| 137 | 20 |
| 138 | 1 |
| 140 | 2 |
| 141 | 1 |
| 143 | 1 |
| 144 | 3 |
| 146 | 3 |
| 147 | 1 |
| 149 | 9 |
| 150 | 1 |
| 152 | 2 |
| 153 | 1 |
| 155 | 1 |
| 156 | 3 |
| 158 | 141 |
| 159 | 32 |
| 161 | 1 |
| 162 | 1 |
| 164 | 1 |
| 165 | 5 |
| 167 | 1 |
| 168 | 1 |
| 170 | 2 |
| 171 | 1851 |
| 173 | 1 |
| 174 | 14 |
| 176 | 1 |
| 177 | 2 |
| 179 | 2 |
| 180 | 2 |
| 182 | 5 |
| 183 | 3 |
| 185 | 12 |
| 186 | 6 |
| 188 | 1 |
| 189 | 1 |
| 191 | 2 |
| 192 | 1 |
| 194 | 1 |
| 195 | 2 |
| 197 | 18 |
| 198 | 2 |
| 200 | 9 |
| 201 | 31274 |
| 203 | 1 |
| 204 | 3 |
| 206 | 1 |
| 207 | 8 |
| 209 | 1 |
| 210 | 27 |
| 212 | 6 |
| 213 | 447 |
| 215 | 1 |
| 216 | 8 |
| 218 | 1 |
| 219 | 2 |
| 221 | 5 |
| 222 | 52725 |
| 224 | 2 |
| 225 | 30 |
| 227 | 36321 |
| 228 | 5 |
| 230 | 2 |
| 231 | 1 |
| 233 | 6 |
| 234 | 2 |
| 236 | 1 |
| 237 | 30 |
| 239 | 2 |
| 240 | 12 |
| 242 | 9 |
| 243 | 2 |
| 245 | 1 |
| 246 | 1 |
| 248 | 3 |
| 249 | 17 |
| 251 | 11 |
| 252 | 53 |
| 254 | 2 |
| 255 | 2 |
| 257 | 662 |
| 258 | 153 |
| 260 | 9 |
| 261 | 32 |
| 263 | 2 |
| 264 | 2 |
| 266 | 1 |
| 267 | 14 |
| 269 | 1398 |
| 270 | 5 |
| 272 | 1 |
| 273 | 102 |
| 275 | 1911 |
| 276 | 12 |
| 278 | 1 |
| 279 | 1 |
| 281 | 8 |
| 282 | 32 |
| 284 | 2 |
| 285 | 2 |
| 287 | 1 |
| 288 | 1 |
| 290 | 1 |
| 291 | 3 |
| 293 | 1 |
| 294 | 2 |
| 296 | 24 |
| 297 | 2 |
| 299 | 26 |
| 300 | 24 |
| 302 | 1085 |
| 303 | 21 |
| 305 | 6 |
| 306 | 2 |
| 308 | 2 |
| 309 | 1 |
| 311 | 6 |
| 312 | 2 |
| 314 | 1 |
| 315 | 48 |
| 317 | 2 |
| 318 | 30 |
| 320 | 5 |
| 321 | 38 |
| 323 | 2 |
| 324 | 3 |
| 326 | 2 |
| 327 | 135981 |
| 329 | 1 |
| 330 | 6 |
| 332 | 1 |
| 333 | 26 |
| 335 | 2 |
| 336 | 1 |
| 338 | 5 |
| 339 | 1002 |
| 341 | 2 |
| 342 | 1 |
| 344 | 1 |
| 345 | 3 |
| 347 | 320 |
| 348 | 1 |
| 350 | 3 |
| 351 | 1 |
| 353 | 3 |
| 354 | 26 |
| 356 | 5 |
| 357 | 1 |
| 359 | 2 |
| 360 | 8 |
| 362 | 200 |
| 363 | 2 |
| 365 | 2 |
| 366 | 5 |
| 368 | 2 |
| 369 | 1 |
| 371 | 5 |
| 372 | 333 |
| 374 | 3 |
| 375 | 2 |
| 377 | 50 |
| 378 | 1 |
| 380 | 24 |
| 381 | 1 |
| 383 | 1 |
| 384 | 2 |
| 386 | 2 |
| 387 | 8 |
| 389 | 2 |
| 390 | 32 |
| 392 | 1 |
| 393 | 17 |
| 395 | 1 |
| 396 | 24 |
| 398 | 1 |
| 399 | 6 |
| 401 | 26 |
| 402 | 1 |
| 404 | 1 |
| 405 | 1 |
| 407 | 2 |
| 408 | 5 |
| 410 | 3 |
| 411 | 35 |
| 413 | 15 |
| 414 | 1 |
| 416 | 1 |
| 417 | 17 |
| 419 | 5 |
| 420 | 9 |
| 422 | 8 |
| 423 | 6 |
| 425 | 11229 |
| 426 | 1 |
| 428 | 2 |
| 429 | 243 |
| 431 | 18 |
| 432 | 2 |
| 434 | 1 |
| 435 | 1 |
| 437 | 8 |
| 438 | 3 |
| 440 | 2 |
| 441 | 12 |
| 443 | 3 |
| 444 | 12 |
| 446 | 20 |
| 447 | 1 |
| 449 | 696 |
| 450 | 2 |
| 452 | 90 |
| 453 | 1 |
| 455 | 1 |
| 456 | 1 |
| 458 | 3 |
| 459 | 9 |
| 461 | 2 |
| 462 | 14 |
| 464 | 18 |
| 465 | 12 |
| 467 | 2 |
| 468 | 9 |
| 470 | 20 |
| 471 | 2 |
| 473 | 3521 |
| 474 | 5 |
| 476 | 1 |
| 477 | 12 |
| 479 | 3 |
| 480 | 2 |
| 482 | 2 |
| 483 | 3 |
| 485 | 12 |
| 486 | 38 |
| 488 | 3 |
| 489 | 1 |
| 491 | 17 |
| 492 | 26 |
| 494 | 56 |
| 495 | 1 |
| 497 | 5 |
| 498 | 15 |
| 500 | 1 |
| 501 | 8 |
| 503 | 2 |
| 504 | 36 |
| 506 | 6 |
| 507 | 200 |
| 509 | 20 |
| 510 | 3 |
| 512 | 1 |
| 513 | 3 |
| 515 | 6 |
| 516 | 9 |
| 518 | 1 |
| 519 | 38 |
| 521 | 498 |
| 522 | 92 |
| 524 | 20 |
| 525 | 1 |
| 527 | 17 |
| 528 | 2 |
| 530 | 1 |
| 531 | 1 |
| 533 | 1 |
| 534 | 6 |
| 536 | 2 |
| 537 | 1 |
| 539 | 6 |
| 540 | 1 |
| 542 | 1806 |
| 543 | 2 |
| 545 | 11 |
| 546 | 2 |
| 548 | 21 |
| 549 | 8 |
| 551 | 1 |
| 552 | 36 |
| 554 | 1 |
| 555 | 2 |
| 557 | 18 |
| 558 | 11 |
| 560 | 1 |
| 561 | 156 |
| 563 | 3 |
| 564 | 2 |
| 566 | 1 |
| 567 | 1 |
| 569 | 96 |
| 570 | 393 |
| 572 | 1 |
| 573 | 50 |
| 575 | 14 |
| 576 | 5 |
| 578 | 450 |
| 579 | 1 |
| 581 | 11 |
| 582 | 1 |
| 584 | 1 |
| 585 | 32 |
| 587 | 8 |
| 588 | 3 |
| 590 | 2 |
| 591 | 2 |
| 593 | 2 |
| 594 | 3 |
| 596 | 5 |
| 597 | 2 |
| 599 | 36 |
| 600 | 8 |
| 602 | 380 |
| 603 | 1 |
| 605 | 1 |
| 606 | 2 |
| 608 | 9 |
| 609 | 1 |
| 611 | 6 |
| 612 | 1 |
| 614 | 11 |
| 615 | 2 |
| 617 | 338 |
| 618 | 2 |
| 620 | 6 |
| 621 | 1 |
| 623 | 3 |
| 624 | 1 |
| 626 | 1 |
| 627 | 6 |
| 629 | 8 |
| 630 | 2 |
| 632 | 18 |
| 633 | 5246 |
| 635 | 1 |
| 636 | 18 |
| 638 | 11 |
| 639 | 3 |
| 641 | 6 |
| 642 | 1 |
| 644 | 1 |
| 645 | 5 |
| 647 | 504 |
| 648 | 114 |
| 650 | 3 |
| 651 | 2 |
| 653 | 6 |
| 654 | 3 |
| 656 | 2 |
| 657 | 14 |
| 659 | 2 |
| 660 | 261 |
| 662 | 50 |
| 663 | 143 |
| 665 | 5 |
| 666 | 3 |
| 668 | 1 |
| 669 | 32 |
| 671 | 1 |
| 672 | 21 |
| 674 | 2 |
| 675 | 2 |
| 677 | 1 |
| 678 | 215 |
| 680 | 2 |
| 681 | 12 |
| 683 | 6774 |
| 684 | 2 |
| 686 | 1 |
| 687 | 2 |
| 689 | 2 |
| 690 | 39 |
| 692 | 2 |
| 693 | 11 |
| 695 | 3 |
| 696 | 1 |
| 698 | 3 |
| 699 | 23 |
| 701 | 1 |
| 702 | 9 |
| 704 | 8 |
| 705 | 38 |
| 707 | 2 |
| 708 | 2 |
| 710 | 24110 |
| 711 | 3 |
| 713 | 2 |
| 714 | 2 |
| 716 | 78 |
| 717 | 37506 |
| 719 | 13418 |
| 720 | 1 |
| 722 | 8 |
| 723 | 6 |
| 725 | 6 |
| 726 | 1 |
| 728 | 1 |
| 729 | 5 |
| 731 | 3 |
| 732 | 50 |
| 734 | 5 |
| 735 | 1 |
| 737 | 36 |
| 738 | 15 |
| 740 | 2 |
| 741 | 2 |
| 743 | 1 |
| 744 | 3 |
| 746 | 2 |
| 747 | 1 |
| 749 | 2 |
| 750 | 2 |
| 752 | 2 |
| 753 | 2 |
| 755 | 1 |
| 756 | 5 |
| 758 | 2 |
| 759 | 2 |
| 761 | 1 |
| 762 | 1 |
| 764 | 20 |
| 765 | 3 |
| 767 | 2 |
| 768 | 1 |
| 770 | 191 |
| 771 | 2 |
| 773 | 1 |
| 774 | 35 |
| 776 | 51 |
| 777 | 2 |
| 779 | 3 |
| 780 | 6 |
| 782 | 1 |
| 783 | 2 |
| 785 | 1 |
| 786 | 5 |
| 788 | 39 |
| 789 | 248 |
| 791 | 27 |
| 792 | 1 |
| 794 | 30 |
| 795 | 5 |
| 797 | 21 |
| 798 | 1 |
| 800 | 8 |
| 801 | 1 |
| 803 | 2 |
| 804 | 6 |
| 806 | 9 |
| 807 | 5 |
| 809 | 5 |
| 810 | 2 |
| 812 | 1 |
| 813 | |
| 815 | 2 |
| 816 | 275 |
| 818 | 1 |
| 819 | 1 |
| 821 | 81 |
| 822 | 2 |
| 824 | 8 |
| 825 | 1 |
| 827 | 1 |
| 828 | 3 |
| 830 | 6 |
| 831 | 9 |
| 833 | 17 |
| 834 | 71 |
| 836 | 1 |
| 837 | 2 |
| 839 | 1 |
| 840 | 29 |
| 842 | 69 |
| 843 | 2 |
| 845 | 2 |
| 846 | 1 |
| 848 | 273 |
| 849 | 9 |
| 851 | 9 |
| 852 | 8 |
| 854 | 3 |
| 855 | 1 |
| 857 | 1 |
| 858 | 3749 |
| 860 | 1 |
| 861 | 2 |
| 863 | |
| 864 | 1 |
| 866 | 2 |
| 867 | 2 |
| 869 | 2 |
| 870 | 3 |
| 872 | 12 |
| 873 | 5 |
| 875 | 1 |
| 876 | 414 |
| 878 | 1 |
| 879 | 11 |
| 881 | 47 |
| 882 | 5 |
| 884 | 5 |
| 885 | 2 |
| 887 | 5 |
| 888 | 1647 |
| 890 | 1 |
| 891 | 816 |
| 893 | 246 |
| 894 | 1 |
| 896 | 8 |
| 897 | 1 |
| 899 | 1 |
| 900 | 2 |
| 902 | 14 |
| 903 | 29 |
| 905 | 6 |
| 906 | 6 |
| 908 | 5 |
| 909 | 164 |
| 911 | 1 |
| 912 | 4917 |
| 914 | 12 |
| 915 | 1 |
| 917 | 2 |
| 918 | 1 |
| 920 | 1 |
| 921 | 137 |
| 923 | 11 |
| 924 | 12 |
| 926 | 6 |
| 927 | 1 |
| 929 | 3615 |
| 930 | 24 |
| 932 | 6 |
| 933 | 2 |
| 935 | 6 |
| 936 | 62 |
| 938 | 17 |
| 939 | 66 |
| 941 | 26 |
| 942 | 5 |
| 944 | 2 |
| 945 | 170 |
| 947 | 2693 |
| 948 | 33 |
| 950 | 1 |
| 951 | 12 |
| 953 | 2 |
| 954 | 2 |
| 956 | 2 |
| 957 | 2 |
| 959 | 1 |
| 960 | 1 |
| 962 | |
| 963 | 357 |
| 965 | 17 |
| 966 | 6 |
| 968 | 5 |
| 969 | 1 |
| 971 | 2 |
| 972 | 26 |
| 974 | 1 |
| 975 | 2 |
| 977 | 24 |
| 978 | 2 |
| 980 | 17 |
| 981 | 1 |
| 983 | 1875 |
| 984 | 17 |
| 986 | 78 |
| 987 | 1 |
| 989 | 2 |
| 990 | 1 |
| 992 | 1 |
| 993 | 1 |
| 995 | 6 |
| 996 | 416 |
| 998 | 2 |
| 999 | 3761 |
| 1001 | 1 |
| 1002 | 1 |
| 1004 | 2 |
| 1005 | 8 |
| 1007 | 1 |
| 1008 | 2771 |
| 1010 | 3351 |
| 1011 | 1 |
| 1013 | 3 |
| 1014 | 3 |
| 1016 | 1 |
| 1017 | |
| 1019 | 89 |
| 1020 | 1 |
| 1022 | 1 |
| 1023 | 6 |
Williams primes
| Main |
| Type MM: (b-1)•bn-1 |
| Type MP: (b-1)•bn+1 |
| Type PM: (b+1)•bn-1 |
| Type PP: (b+1)•bn+1 |
