In the original Sierpinski (base 2) problem, the conjectured smallest Sierpinski number is 39565, 39565×2n+1 is divisible by at least one of the primes 3, 5, 7, 11, 17, 31, 61 for all n≥1, and there are 5 odd k below 39565 remaining with no known prime of the form k×2n+1:

10311, 11177, 12395, 28117, 3315E

(also 1 even k such that k+1 is prime, but without known prime for n≥1: 31E14)

The largest 10 known primes:

 k n Dozenal length of the prime 5XEE X533545 2XE3E90 E181 4439XX2 1271647 14005 30X1289 X33X37 14555 2757621 894053 17591 2431000 7X7031 3127 1839086 57EE16 297E 1141XE3 388133 2783E 545X87 15EX60 33EE1 47999X 136X16 31E3E 40X837 1177X6 21657 400458 114940 22865 29807E 94865

A Colbert number is a prime >1000000 digits (i.e. ≥101000000) and has contributed to the in-progress computational proof that 39565 is the smallest Sierpinski number, there are only two known such numbers: E181×24439XX2+1 (1271647 digits) and 5XEE×2X533545+1 (2XE3E90 digits), similarly, a Colbert number base b is a prime >1000000 digits (i.e. ≥101000000) and has contributed to the in-progress computational proof the smallest generalized Sierpinski number base b (numbers k such that k×bn+1 is composite for all n≥1 and gcd(k+1,b−1) = 1 and k is not rational power of b (or this formula would be corresponding to generalized Fermat numbers)).

In the original Riesel (base 2) problem, the conjectured smallest Riesel number is 206817, 206817×2n−1 is divisible by at least one of the primes 3, 5, 7, 11, 15, 181 for all n≥1, and there are 41 odd k below 206817 remaining with no known prime of the form k×2n−1:

13E1, 5405, 11845, 1652E, 1X321, 23007, 32X11, 3728E, 3XX95, 4637E, 4826E, 52157, 5X655, 627X7, 6X947, 70995, 79685, 9380E, 9E29E, 9E41E, X4817, XXX61, E3747, EX135, 100837, 1329E5, 134X9E, 13760E, 138197, 13975E, 142E1E, 1464X7, 14761E, 155X15, 156327, 157207, 15739E, 15924E, 15E271, 16664E, 167X15, 171917, 179161, 195391, 1X8111, 1XX70E, 1E0467, 1E4EE1, 1EX387

(also 2 even k such that k−1 is prime, but without known prime for n≥1: 14E252 and 1E08E2)

The largest 10 known primes:

 k n Dozenal length of the prime 112555 2EX9280 X01E08 202X11 24X42XE 807436 174E4E 249E080 805XE2 1E5E1 2344139 77709X 128067 22847X5 754539 172407 21961X3 72405X 101E3E 1X11786 61E600 15045E 154X524 4X31X4 6X185 1534126 49X05E 5E5E7 1335X35 432242 180397 131X9E5 428X7E 31E0E 1270392 409X54

## Sierpinski 2nd problem

The conjectured 2nd Sierpinski number is 110XX1, and there are 14 odd k between 39565 and 110XX1 remaining with no known prime of the form k×2n+1:

39X91, 3X235, 44E91, 63XE7, 7414E, 766X7, 7X52E, 98211, 99381, X1377, X8655, XX8E7, XE94E, E0XE5, E51E7, E5E77

(also 2 even k such that k+1 is prime, but without known prime for n≥1: 4455X and 4XX6X)

The largest 10 known primes:

 k n Dozenal length of the prime 81597 65X4600 1987799 49877 4840XX6 138705E 94325 3X039E7 10X0959 4447E 309X21E X33013 79241 24694X4 7E74XX 1055X5 19XX29E 613X31 109927 1763420 554261 X2277 10E637X 374719 46745 10X78X5 371794 96E85 EX3583 33835E X8821 E58134 324X1E 71487 X36922 2X573X

## Dual Sierpinski problem

All odd k below 39565 have at least one known (probable) prime of the form 2n+k.

The largest (probable) prime is 23065974+1E397 (for k = 1E397), it has X238XE digits.

The largest 10 (probable) primes:

 k n Dozenal length of the (probable) prime 1E397 3065974 X238XE 20165 188221E 5928XE 1297 1650374 517X11 14555 90579E 263106 37735 6126EE 1850X9 4E3E 495553 14039X 2XE97 3E5284 112946 E181 22721X 75049 19E7E 12X861 41X25 31145 127E27 410XE 2E32E 85E55 24534 2XE8E 71943 1EE17

For 14 values of k, only non-proven strong probable primes are known (the largest proven prime is EE07-digit 236899+8625, and the smallest non-proven strong probable prime is 14901-digit 250072+X277, the current limit of primality proof (other than the numbers N such that either N−1 or N+1 has ≥40% factored) is 10000 digits).

## Dual Riesel problem

Odd k below 206817 without known (probable) primes of the form 2nk

There are two definitions: allow negative primes or not.

## Table of "Sierpinski, Riesel, dual Sierpinski, dual Riesel" problem for odd k up to 1000

 k Sierpinski (smallest n≥1 such that k×2n+1 is prime) Riesel (smallest n≥1 such that k×2n−1 is prime) dual Sierpinski (smallest n≥1 such that 2n+k is prime) dual Riesel (smallest n≥1 such that |2n−k| is prime) 1 1 2 1 2 3 1 1 1 3 5 1 2 1 1 7 2 1 2 1 9 1 1 1 1 E 1 2 1 2 11 2 3 2 1 13 1 1 1 1 15 3 2 1 2 17 6 1 2 1 19 1 1 1 1 1E 1 4 3 2 21 2 3 2 1 23 2 1 1 2 25 1 4 1 4 27 8 1 4 1 29 1 2 2 1 2E 1 2 1 2 31 2 1 2 3 33 1 3 1 1 35 1 2 1 2 37 2 7 2 1 39 2 1 1 1 3E 407 4 5 2 41 2 1 2 1 43 1 1 1 2 45 1 2 3 4 47 4 1 2 1 49 2 1 1 2 4E 5 10 1 4 51 4 3 8 1 53 1 2 2 1 55 1 4 1 2 57 2 5 2 3 59 1 1 1 1 5E 3 2 1 2 61 2 7 4 1 63 1 1 2 1 65 3 2 1 2 67 2 1 2 3 69 1 3 1 1 6E 1 2 7 2 71 4 5 2 1 73 2 1 1 2 75 1 4 3 4 77 8 1 4 1 79 2 3 2 2 7E 1 2 1 4 81 2 1 2 3 83 1 1 1 1 85 3 X 1 2 87 14 3 2 1 89 1 2 1 1 8E 3 X 1 2 91 6 9 2 1 93 1 2 1 1 95 1 8 7 2 97 2 1 4 1 99 3 1 5 2 9E 1 10 3 4 X1 8 1 4 3 X3 6 2 2 4 X5 1 2 1 4 X7 2 21 2 3E X9 3 1 1 1 XE 1 2 3 2 E1 4 3 2 1 E3 1 1 1 2 E5 3 2 1 6 E7 2 1 X 1 E9 1 1 3 1 EE 45 2 3 2 101 6 5 2 3 103 8 1 1 3 105 3 4 1 8 107 4 5 4 1 109 1 3 2 1 10E 1 2 1 2 111 8 1 4 3 113 6 1 2 1 115 3 2 1 2 117 2 3 2 5 119 1 2 1 1 11E 7 4 5 2 121 2 1 2 1 123 8 2 1 2 125 1 2 3 4 127 2 1 2 1 129 2 1 1 2 12E 1 8 1 4 131 4 3 4 1 133 1 4 3 1 135 3 2 3 2 137 6 1 2 3 139 1 3 1 3 13E 1 16X 1 6 141 2 3 2 1 143 4 1 1 1 145 13 2 1 2 147 2 1 6 1 149 3 1 5 1 14E 11 2 3 2 151 2 1 6 3 153 2 2 2 3 155 1 4 1 4 157 18 1 4 5 159 2 4 4 1 15E 1 2 3 2 161 56 1 4 7 163 1 7 2 3 165 1 2 1 6 167 8 3 2 5 169 3 1 1 1 16E E 2 1 2 171 6 1 2 1 173 1 1 1 1 175 1 4 3 2 177 2 3 2 1 179 4 2 1 2 17E 1 4 1 4 181 30 E 4 1 183 1 2 3 1 185 1 4 5 2 187 6 E1 2 3 189 1 8 1 3 18E 1 6 5 8 191 2 5E 2 1 193 2 1 1 2 195 1E3 4 9 4 197 32 5 2 1 199 1 1 1 2 19E 25 2 3 6 1X1 2 E 2 1 1X3 2 4 1 2 1X5 3 8 1 8 1X7 4 1 18 1 1X9 1 2 2 1 1XE 7 6 1 2 1E1 2 5 2 3 1E3 2 1 1 1 1E5 1 2 1 2 1E7 26 E 6 1 1E9 1 1 3 1 1EE 3 4 25 2 201 X 1 2 3 203 4 2 1 3 205 1 2 7 4 207 2 5 4 1 209 3 1 4 2 20E 1 4 3 4 211 4 1 4 3 213 1 3 2 5 215 3 8 1 6 217 2 1 2 7 219 1 1 1 1 21E 9 22 1 2 221 4 3 2 1 223 1 2 1 1 225 7 6 5 2 227 2 3 6 1 229 1 1 4 2 22E 1 2 3 4 231 2 5 6 3 233 3 1 2 4 235 1 8 1 4 237 4 1 4 9 239 5 3 2 1 23E 17 16 1 2 241 4 1 4 55E 243 3 1 3 1 245 1 X 3 2 247 2 1E 2 5 249 1 4 1 3 24E 3 16 1 4 251 X 3 2 1 253 10 1 1 1 255 19 4 23 2 257 6 1 2 1 259 2 2 1 2 25E 1 4 3 8 261 24 3 8 1 263 1 2 2 2 265 5 2 1 4 267 10 1 4 3 269 1 4 2 1 26E 1 2 1 2 271 2 7 4 9 273 1 2 2 1 275 E X 1 2 277 12 1 2 5 279 3 1 1 1 27E 3849 2 13 2 281 8 1 2 1 283 2 1 1 2 285 E 18 3 4 287 4 5 8 1 289 1 2 2 2 28E 5 2 1 4 291 4 5 2 3 293 2 1 1 1 295 3 X 3 2 297 2 E 4 1 299 1 1 2 2 29E 3 2 1 8 2X1 2 3 X 3 2X3 1 1 3 1 2X5 1 4 3 2 2X7 6 1 2 5 2X9 2 2 1 3 2XE 1 4 1 8 2E1 4 7 8 1 2E3 2 8 3 1 2E5 7 2 3 2 2E7 2 1 2 3 2E9 1 1 1 3 2EE 1 2 1 6 301 2 3 4 1 303 2 4 2 1 305 1E 2 1 2 307 6 1 2 3 309 1 1 1 1 30E 1 6 7 2 311 4 3 2 1 313 2 2 1 2 315 3 94 3 4 317 10 3 4 1 319 1 2 2 2 31E 1 18 1 4 321 8 9 2 3 323 1 3 1 1 325 5 X 1 2 327 6 7 2 1 329 2 1 1 1 32E 17 2 5 2 331 2 1 X 1 333 3 1 3 2 335 1 6 7 4 337 2 5 2 3 339 6 1 1 4 33E 3 10 3 4 341 54 3 18 1 343 1 2 2 2 345 1 4 1 6 347 2 5 2 3 349 5 1 1 1 34E 1 3X 3 2 351 2 3 4 1 353 1 2 2 2 355 7 2 1 6 357 2 1 2 3 359 4 2 1 1 35E 5 2 7 2 361 4 1 2 1 363 2 1 1 2 365 1 8 5 9 367 8 1 14 1 369 2 6 3 2 36E 1 8 3 4 371 2 1 2 3 373 1 5 1 4 375 5 2 1 6 377 4 7 6 1 379 1 1 4 1 37E 3 4 9 2 381 6 3 6 3 383 1 1 4 3 385 11 2 3 8 387 2 1 6 5 389 3 5 2 4 38E 7 4 1 4 391 8 3 4 5 393 1 13 2 1 395 1 2 1 2 397 4 1 4 7 399 5 1 3 1 39E 1 2 9 2 3X1 2 3 2 5 3X3 2 1 1 3 3X5 3 8 9 4 3X7 2 1 2 1 3X9 1 2 1 2 3XE 9 2 3 4 3E1 4 1 2 1 3E3 2 2 1 2 3E5 13 4 1 8 3E7 4 3 4 1 3E9 2 3 2 1 3EE 1 4 1 2 401 2 1 4 3 403 6 E 3 1 405 1 16 5 2 407 2 3 2 9 409 1 2 1 3 40E 16E 2 5 4 411 2 7 2 1 413 3 1 1 2 415 1 2 3 4 417 2 3 2 1 419 2 1 1 2 41E 3 8 1 14 421 14 1 4 1 423 4 2 2 1 425 E 4 1 2 427 10 1 8 3 429 2 1 2 1 42E 1 6 1 2 431 6 3 2 9 433 1 1 1 1 435 3 2 1 2 437 2 1 6 1 439 3 3 5 1 43E 5 4 3 2 441 8 1 4 3 443 6 5 2 3 445 1 14 1 4 447 100 15 4 5 449 9 2 3 1 44E 3 2 3 2 451 2 9 2 7 453 1 1 1 3 455 1 6 1 6 457 4 3E 2 1 459 1 1 1 1 45E 13 X 11 2 461 1X 1 2 1 463 1 1 1 2 465 1 X 3 8 467 2 5 2 1 469 4 5 1 2 46E 1 327318 1 4 471 8 1 4 1 473 1 3 6 1 475 7 2 3 2 477 6 59 4 3 479 2 3 2 3 47E 7 2 1 6 481 2 7 2 5 483 5 2 1 1 485 3 2 5 2 487 6 3 2 1 489 3 1 1 2 48E 1 2 3 6 491 2 3 4 1 493 2 1 2 2 495 13 8 1 4 497 4 1 8 3 499 8 4 3 1 49E 9 4 5 2 4X1 2 5 2 7 4X3 1 3 1 3 4X5 5 2 3 12 4X7 8 3 4 1 4X9 3 1 2 2 4XE 3 4 1 4 4E1 2 59 6 3 4E3 1 2 3 1 4E5 1 2 17 2 4E7 2 1 2 5 4E9 3 1 1 3 4EE 1 4 3 8 501 8 5 8 1 503 1 3 2 2 505 1 6 1 4 507 2 1 4 3 509 1 5 2 1 50E 3 12 1 2 511 10 17 6 5 513 1 2 2 1 515 3 50 1 2 517 2 8E 2 9 519 1 1 1 1 51E 1 2 3 2 521 6 1 4 1 523 4 1 2 2 525 1 10 1 4 527 10 3 20 3 529 4 2 2 1 52E 1 2 1 2 531 4 5 2 E 533 2 4 1 1 535 1 1X 3 2 537 8 47 6 1 539 1 4 2 2 53E 1E 2 1 4 541 12 3 2 3 543 1 2 1 1 545 45 8 677 2 547 4 1 6 1 549 2 1 5 2 54E 1 14 3 8 551 4 3 4 3 553 1 3 2 5 555 1 44 1 4 557 8 5 X 7 559 1 3 3 1 55E 1 2 5 2 561 4 3 2 5 563 2 3 1 3 565 2E 2 5 6 567 6 1 6 1 569 5 1 3 2 56E 1 10 3 4 571 2 1 2 3 573 2 1 1 6 575 1 10 1 450 577 14 1 4 1 579 1 2 3 1 57E 3 2 3 2 581 6 9 2 3 583 3 1 1 3 585 3 6 1 6 587 6 E 2 1 589 2 2 1 1 58E 17 2 1 2 591 16 1 26 1 593 1 2 3 1 595 1 2 13 2 597 6 1 2 3 599 6 2 1 3 59E 5 8 7 4 5X1 4 7 4 1 5X3 2 2 4 2 5X5 3 8 3 4 5X7 2 1 4 3 5X9 1 1 2 8 5XE 7 12 1 6 5E1 2 3 2 5 5E3 3 1 1 1 5E5 3 4 1 2 5E7 X 3 2 1 5E9 1 1 1 1 5EE 5 4 7 2 601 2 9 4 1 603 2 1 4 2 605 5 4 3 4 607 8 1 4 3 609 1 2 2 4 60E 3 2 1 4 611 4 1 2 E 613 1 8 1 1 615 717 6 1 2 617 2 7 2 1 619 2 2 1 1 61E 7 2 5 2 621 2 1 6 1 623 1 3 4 2 625 1 2 7 4 627 2 1 4 3 629 3 5 5 4 62E 3 8 3 4 631 10 1 90 7 633 2 4 2 4 635 1 4 1 10 637 6 5 2 203 639 2 4 1 1 63E 1 2 3 2 641 16 E 4 1 643 1 2 2 2 645 23 6 1 6 647 2 3 6 3 649 3 6 3 1 64E 1 2 7 2 651 2 5 2 7 653 2 4 1 3 655 3 10 3 8 657 4 1 4 1 659 1 X 2 2 65E 1 2 1 4 661 8 1 2 3 663 1 1 1 1 665 3 6 7 2 667 6 1E 2 1 669 3 1 1 2 66E 3 6 9 6 671 2 3 2 1 673 4 1 1 2 675 1 12 E 4 677 2 3 4 1 679 4 1 6 2 67E 3 18 3 10 681 4 3 4 3 683 2 2 2 4 685 1 4 1 8 687 4 1 2 7 689 2 8 1 1 68E 9 6 9 2 691 4 7 2 1 693 1 1 1 2 695 3 2 5 X 697 2 5 2 1 699 5 2 1 2 69E 9 2 3 4 6X1 4 3 6 1 6X3 7 1 2 2 6X5 1 4 1 8 6X7 14 3 10 3 6X9 1 4 2 1 6XE 9 4 1 2 6E1 2 1 4 E 6E3 1 1 5 1 6E5 1 2 3 2 6E7 2 13 4 5 6E9 1 2 2 3 6EE 1E 2 1 4 701 6 1 2 5 703 3 3 1 1 705 1 2 3 2 707 8 1 2 1 709 4 9 1 2 70E 1 8 1 X 711 94 3 8 1 713 2 2 3 1 715 5 2 3 2 717 4 1 2 3 719 3 3 1 3 71E 1 6 1 6 721 2 3 4 1 723 4 1 2 1 725 3 8 1 2 727 2 3 6 3 729 1 1 3 1 72E 1 6 3 2 731 8 1 2 5 733 3 5 1 3 735 1 10 1 4 737 8 11 8 1 739 12 2 3 1 73E 1 2 3 2 741 2 1 2 3 743 5 4 1 3 745 9 2 1 6 747 2 27 8 1 749 1 1 2 1 74E 3 14 1 2 751 X 1 16 3 753 1 1 4 1 755 5 4 7 2 757 8 3 4 7 759 17 1 4 3 75E 7 14 3 4 761 10 1 4 5 763 6 3 2 5 765 3 2 1 4 767 2 21 2 2E 769 1 5 1 1 76E 5 2 1 2 771 2 27 2 1 773 3 4 1 1 775 7 4 5 2 777 2 19 2 1 779 1 3 1 2 77E 1 2672 7 4 781 2 3 2 1 783 4 1 1 2 785 35 18 3 4 787 10 1 8 1 789 5 2 2 2 78E 9 4 1 6 791 8 9 6 3 793 1 1 2 1 795 1 2 1 2 797 2 7 6 5 799 1 3 2 1 79E 13 2 1 2 7X1 2 7 6 5 7X3 3 2 5 1 7X5 1 4 E 2 7X7 30 1 4 5 7X9 2 1 4 3 7XE 9 4 5 4 7E1 4 1 10 5 7E3 1 4 3 8 7E5 3 12 3 4 7E7 4 1 2 7 7E9 2 1 1 5 7EE 3 2 1 6 801 16 7 6 1 803 1 1 3 1 805 3 6 9 2 807 2 5 2 3 809 7 2 1 3 80E 5 2 3 8 811 6 3 4 1 813 3 1 2 2 815 1 14 1 4 817 8 1 4 3 819 1 2 3 1 81E 1 8 7 2 821 6 5 2 7 823 3 1 1 3 825 7 2 5 6 827 2 3 2 1 829 1 3 1 2 82E 3 6 9 4 831 X 1 2 1 833 1 1 1 2 835 5 4 3 6 837 4 1 6 1 839 2 1 2 2 83E 1 4 1 10 841 680 5 4 3 843 2 3 8 1 845 1 1X 3 2 847 6 9 4 4E 849 6 1 2 3 84E 1 12736 1 12 851 28 17 2 5 853 2 3 1 1 855 7 4 5 2 857 2 1 2 1 859 3 1 1 2 85E 1 8 3 8 861 6 5 2 1 863 2 10 1 2 865 1 8 1 4 867 4 11 8 1 869 1 2 2 1 86E 5 4 1 2 871 4 1 6 3 873 2 1 6 1 875 3 6 3 2 877 2 23 4 11 879 5 4 2 3 87E 7 2 1 4 881 12 7 X 5 883 1 2 3 1 885 15 2 7 2 887 2 3 2 5 889 4 6 1 3 88E 5 4 5 10 891 4 1 4 1 893 6 2 4 2 895 1 2 5 4 897 10 5 4 3 899 1 3 3 5 89E 1 6 3 16 8X1 4 13 2 9 8X3 1 1 1 4 8X5 EE 2 1 6 8X7 1X 1 2 1 8X9 5 3 1 1 8XE 5 10 3 2 8E1 6 E 2 1 8E3 4 2 1 2 8E5 1 14 1 8 8E7 10 7 4 1 8E9 X 2 2 1 8EE 1 2 1 2 901 2 1 2 3 903 2 3 1 1 905 E 12 1 2 907 4 7 2 1 909 5 1 1 1 90E 3 2 39 2 911 2 1 6 1 913 5 1 3 2 915 5 22 3 4 917 2 15 2 3 919 8 1 1 4 91E 5 10 1 4 921 10 3 10 1 923 1 7 2 1 925 3 60 1 2 927 2 5 8 3 929 1 1 5 1 92E 1 2 7 2 931 2 3 X 5 933 1 5 5 3 935 13 2 15 4 937 X 1 26 5 939 1 12 5 6 93E 1 4 7 4 941 2 1 4 7 943 6 1 X 6 945 1 38 5 8 947 4 13 4 5 949 1 4 3 5 94E 1 2 7 6 951 4 1 2 7 953 1 5 1 5 955 3 2 7 6 957 14 3 2 1 959 1 1 1 2 95E 3 4 5 6 961 2 19 2 1 963 4 1 1 2 965 39 24 3 8 967 2 1 6 1 969 6 1 2 2 96E 5 4 1 8 971 10 3 8 3 973 1 2 4 1 975 E 4 E 2 977 4 5 6 7 979 2 1 6 3 97E 33 2 3 6 981 2 17 4 5 983 1 1 2 9 985 3 10 1 4 987 X 1 X 5 989 1 1 3 1 98E 9 4 7 2 991 4 3 2 5 993 3 4 1 3 995 1 4 E 8 997 8 3 4 1 999 2 2 4 2 99E 3 2 3 4 9X1 2 1 4 3 9X3 3 1 2 7 9X5 1 2 1 X 9X7 2 3 2 9 9X9 1 3 1 1 9XE 7 14 1 2 9E1 2 1 2 1 9E3 9 1 1 1 9E5 9 6 33 2 9E7 2 21 2 1 9E9 2 6 1 2 9EE 1 10 3 4 X01 4 3 8 1 X03 1 4 2 2 X05 13 2 1 4 X07 5X E1 2 3 X09 3 1 1 1 X0E 1 6 1 2 X11 2 2E 8 1 X13 2 1 2 1 X15 3 2 1 2 X17 6 1 6 3 X19 3 2 5 1 X1E 1 2 3 2 X21 2 3 4 5 X23 2 2 2 3 X25 1 8 1 4 X27 4 11 4 5 X29 5 3 3 1 X2E 3 4 3 2 X31 4 1 2 E X33 3 1 1 3 X35 1 2 1 X X37 8 3 2 1 X39 1 1 1 1 X3E 3 10 1 2 X41 16 7 2 1 X43 4 3 1 1 X45 5 4 7 2 X47 2 3 2 1 X49 X 2 1 2 X4E 1 14 5 4 X51 8 1 24 1 X53 4 2 3 2 X55 1 8 9 4 X57 2 9 2 3 X59 1 3 1 4 X5E 1 2 5 6 X61 2 7 14 1 X63 5 3 3 2 X65 13 2 5 6 X67 2X 1 2 3 X69 4 1 1 5 X6E 101 2 3 6 X71 2 1 X 1 X73 6 11 2 2 X75 9 14 1 14 X77 14 1 8 3 X79 1 2 4 1 X7E 3 X 3 2 X81 4 19 4 7 X83 1 4 2 3 X85 1 2 1 12 X87 2 3 4 5 X89 X 1 2 1 X8E 3 8 1 2 X91 2 3 2 E X93 3 2 1 1 X95 5 2 27 2 X97 2 1 2 1 X99 2 5 1 2 X9E 1 4 3 4 XX1 4 1 10 1 XX3 10 3 2 2 XX5 11 16 1 4 XX7 2 49 2 3 XX9 2 1 1 1 XXE 3 X 3 2 XE1 4 3 6 1 XE3 2 2 2 2 XE5 2E 6 1 8 XE7 2 5 2 3 XE9 1 2 1 1 XEE 1 8 E 2 E01 12 1 4 1 E03 3 4 5 2 E05 3 4 3 50 E07 4 1 4 3 E09 1 3 2 X E0E 1 2 1 4 E11 2 11 2 40E E13 2 3 1 1 E15 1 12 3 2 E17 6 3 2 1 E19 2 1 1 2 E1E E 2 1 6 E21 X 1 2 1 E23 3 1 1 1 E25 9 2 3 2 E27 4 1 2 1 E29 2 8 1 2 E2E 3 4 1 10 E31 8 3 4 1 E33 4 2 2 1 E35 1 4 1 2 E37 8 1 X6 3 E39 1 1 3 1 E3E 3 6 5 2 E41 6 3 2 5 E43 1 6 1 3 E45 27 2 5 4 E47 X 15 X 1 E49 8 2 4 2 E4E X21 2 E 4 E51 2 3 6 3 E53 3 6 4 6 E55 1 40 3 10 E57 4 1 4 5 E59 1 7 2 4 E5E 3 2 1 8 E61 4 1 6 33 E63 1 5 2 1 E65 1 6 1 2 E67 2 E 2 9 E69 1 1 1 1 E6E 3 24 1 2 E71 172 5 6 1 E73 1 3 E 1 E75 1 2 7 2 E77 2 3 8 3 E79 2 9 4 3 E7E 1 4 E 4 E81 10 1 4 7 E83 2 3 4 4 E85 1 4 3 4 E87 4 1 X 7 E89 3 13 2 5 E8E 31 2 1 6 E91 14 7 2 9 E93 1 1 1 1 E95 3 8 1 2 E97 2E6 3 16 1 E99 13 2 3 1 E9E 1 6 7 2 EX1 4 5 2 3 EX3 2 1 1 3 EX5 E 14 5 4 EX7 8 3 10 1 EX9 4 4 3 2 EXE 3 14 3 4 EE1 2 1 2 3 EE3 5 3 1 10 EE5 5 2 1 6 EE7 6 7 6 1 EE9 5 1 3 1 EEE 267 2 5 2

## Brier number

Numbers k which is both Sierpinski and Riesel.

CK=E0739280X466X9X4771997671

All (odd and even) k ≤ 1010 have at least one known prime of the form either k×2n+1 or k×2n−1, with n≥1

## The Liskovets-Gallot Conjectures

Smallest odd k divisible by 3 (i.e. k ends with either 3 or 9) such that k×2n+1 or k×2n−1 is not prime for all even n or all odd n

In the "Riesel with even n" case, square k are excluded, since k×2n−1 has algebra factors for all even n

### Sierpinski with even n

CK=32759, proven (only the k end with 3 or 9 are considered, the k end with 6 are "Sierpinski with odd n" for k/2, and the k end with 0 are "Sierpinski with even n" or "Sierpinski with odd n" for larger n, like the Sierpinski/Riesel conjectures base b for the k which is multiples of b)

The largest 10 primes:

 k n Dozenal length of the prime 116X3 1303EE8 423774 2E269 103E4E6 3532XX 20879 378228 102279 17733 16291X 51037 17663 87160 24928 11933 51342 1511X 25743 462E4 13174 8599 44X60 12903 20179 3X782 1101X 2X033 39E64 109X7 10683 3620X E924 30843 35854 E777

### Sierpinski with odd n

CK=47183, 3 k remain: 5443, 167E3, 268E9 (only the k end with 3 or 9 are considered, the k end with 6 are "Sierpinski with even n" for k/2, and the k end with 0 are "Sierpinski with even n" or "Sierpinski with odd n" for larger n, like the Sierpinski/Riesel conjectures base b for the k which is multiples of b)

The largest 10 known primes:

 k n Dozenal length of the prime 409X3 8E2095 25X8XX 41433 77190E 215122 2XE19 68X523 1X6846 3X693 1X6XE9 636X6 12359 152X09 49848 19E69 124E95 40308 34943 10E473 3741X 1X8X9 82107 23444 2E469 7X387 22387 44XE9 7963E 22112 25229 76759 21346 1853 52635 15539

### Riesel with even n

CK=1E143, 2 k remain: 5613, 8389 (only the k end with 3 or 9 are considered, the k end with 6 are "Riesel with odd n" for k/2, and the k end with 0 are "Riesel with even n" or "Riesel with odd n" for larger n, like the Sierpinski/Riesel conjectures base b for the k which is multiples of b)

The largest 10 known primes:

 k n Dozenal length of the prime E289 104X182 356298 E729 694504 1X8450 13413 80896 22E9E 15563 38192 10398 9073 32474 X862 13489 2291X 7575 1319 19E16 6151 6703 167E8 5261 5229 15XE2 4EE9 165E3 E4X0 3225 16233 997X 28X3 8723 65E8 1994

### Riesel with odd n

CK=83E19, 4 k remain: 1XE73, 501X3, 75379, 7X753 (only the k end with 3 or 9 are considered, the k end with 6 are "Riesel with even n" for k/2, and the k end with 0 are "Riesel with even n" or "Riesel with odd n" for larger n, like the Sierpinski/Riesel conjectures base b for the k which is multiples of b)

The largest 10 known primes:

 k n Dozenal length of the prime 76259 9177E5 266EE8 71X03 7E1E73 226669 71573 3482E5 E4242 65649 331739 XXEE7 4013 2EX309 X0029 15443 25701E 83046 51689 1XE269 64932 70089 1944E1 5E63X 42159 1526E3 49763 4X13 131661 42786 83773 1176X1 39768 63603 1122X5 38187

## Smallest odd k divisible by 3 (i.e. k ends with either 3 or 9) such that k×2n±1 are not both primes for all n≥1

CK=179, 9 k remain: 93, X3, 109, 113, 123, 133, 139, 163, 169 (only the k end with 3 or 9 are considered)

## Smallest odd k divisible by 3 (i.e. k ends with either 3 or 9) such that k×2n+1 or k×2n−1 is not both primes for two consecutive n≥1

In the "Riesel" case, square k are excluded, since k×2n−1 has algebra factors for all even n

### Sierpinski

CK=16959, 13X6 k remain, the smallest remain k are 43, 73, 79, 93, X3, X9, 159, 1E3, ... (only the k end with 3 or 9 are considered)

### Riesel

CK=573, 28 k remain: 33, 133, 159, 163, 1X9, 1E3, 239, 253, 283, 289, 293, 2X9, 2E9, 333, 383, 393, 3X9, 3E3, 3E9, 413, 419, 423, 453, 489, 493, 499, 4X3, 503, 529, 533, 543, 563 (only the k end with 3 or 9 are considered)

## Sierpinski problem base b

### Table for bases up to 20

All n must be ≥ 1.

k-values with at least one of the following conditions are excluded from the conjectures:

1. All n-values have a single trivial factor.

2. Make a full covering set with all or partial algebraic factors.

3. Make generalized Fermat numbers, i.e. qm×bn+1 where b is the base, m≥0, and q is a root of the base.

k-values that are a multiple of base (b) and where k+1 is composite are included in the conjectures but excluded from testing. Such k-values will have the same prime as k / b.

 base conjectured smallest Sierpinski number covering set ks that make a full covering set with all or partial algebraic factors trivial ks number of remaining k remaining k top 6 primes comments 2 39565 {3, 5, 7, 11, 17, 31, 61} (none) 5 10311, 11177, 12395, 28117, 3315E 5XEE×2X533545+1E181×24439XX2+114005×230X1289+114555×22757621+117591×22431000+13127×21839086+1 k = 31E14 is GFN with no known prime. 3 2029E39E01X {5, 7, 11, 15, 17, 31, 35, 141, 531} k = = 1 mod 2 (2) 314419 216X6X4, 2E98X24, 42X8792, 489X108, 4EE752X, 6167974, 72489XX, 8004E24, 8064188, 83210E4, E162E22, E21998X, EEE5XX4, 104E494X, 11257454, 13854068, 13E4X804, 1630552X, 17071EX4, 18356878, 18629454, 19278148, 1E33047X, 1E5X85XX, 1E642E88, 20718458, 215611E8, 21E4357X, 21E90XE8, 22435EX4, 231XX22X, 235X721X, 25602752, 2736E082, 28227482, 2887555X, 2943X62X, 2X251398, 31194428, 341689E8, 34237664, 352030XX, 358959X2, 35E24E0X, 36114618, 36662124, 38708X0X, 3903966X, 3X62XE28, 3XE55778, 3E754184, 40709444, 41721644, 41801E88, 4194X4X8, 42X64782, 434X2192, 4398E032, 44712458, 453E2EX4, 46E38214, 47540398, 49316222, 4X56667X, 4EX20X7X, 4EXX8572, 5056348X, 50578448, 518E4774, 5227X654, 52390828, 534071X0, 54093214, 54371708, 54715658, 547E5342, 54E29758, 5694X484, 56XX86XX, 57357838, 58X1X4X8, 59465X78, 5X466062, 5E182E94, 5EX45598, 61618288, 61942586, 62X76294, 6303495X, 640XE0XX, 64496568, 64E49664, 64E8107X, 6505X962, 65436434, 65E16X92, 67X535EX, 693E605X, 6944E64X, 699X950X, 6X23X708, 6X405E6X, 6X7XX8X2, 6X983E34, 70297378, 706X69E8, 70X1X202, 722X9X4X, 7422X958, 7452326X, 74860882, 75X35352, 76057662, 78934028, 7X392212, 7X581788, 8270814X, 843288E2, 850506E2, 85218542, 860484EX, 863753E8, 87000418, 89539E7X, 90XX074X, 9171E9X4, 9267X818, 9348X6X2, 93X59952, 94603676, 95190314, 95920288, 9598E694, 96791X1X, 97327X32, 9780801X, 9914X554, 9987XE52, 9X024X78, X1851268, X3035712, X315182X, X3928374, X39490X2, X40633E2, X408E7EX, X4892X34, X5X6X18X, X6575152, X6X08534, X6X2X094, X8371084, X9127902, XE7E12E4, XE835352, XEX41564, E0414172, E095X638, E1781672, E204X314, E2128524, E221E2X2, E33X4312, E39E0E16, E40E9664, E5542404, E5907072, E6291E38, E7598X74, E7618E7X, E8355102, E9263064, E9971438, EX28894X, EX5XX652, EE58E744, ..., 2028E621292, 2029183X27X, 20292148X02, 20292405058, 2029330905X, 20294138908, 202945512E2, 20295474368, 20295635874, 20296150E12, 20297828E44, 20299X8970X 14E97X938×32002EX+122X1576E2×31EX80E+128E4EE5X×31EX258+13X5X184X×31E83E2+122E32X274×31E65E5+1E585E204×31E4972+1 4 32759 {5, 7, 11, 15, 181} k = = 2 mod 3 (3) 6 X886, 10311, 11177, 2476X, 28117, 313X6 E9EX×45277882+1E181×4221XE51+12800X×41650744+128XXX×41389910+117591×41216600+13127×4X1X643+1 k = 31E14 is GFN with no known prime. 5 78702 {3, 7, 11, 27, 421} k = = 1 mod 2 (2) 27 3884, 4434, 639X, 13612, 1538X, 16428, 190X4, 201X2, 217E8, 21X82, 22584, 25774, 29E2X, 2XE4X, 3034X, 3122X, 33164, 33258, 35458, 37234, 38498, 406X8, 40938, 44148, 45922, 46124, 51048, 58748, 60EE2, 68678, 74378 681XX×5E18156+13E244×5X25X74+1453EX×5875400+138728×5872469+1752EX×584X3X0+16E444×5813EXX+1 6 84X58 {7, 11, 27, 31, 81} k = = 4 mod 5 (5) X 7793, 8489, 250E8, 382X1, 42988, 48263, 55327, 61X88, 80601, 816XX 5E9E9×6813E7X+168819×65188X0+11760X×637XX7X+16048X×6372E08+116178×6342148+12X52X×6317X91+1 k = 900, 4600, and 23000 are GFN with no known prime. 7 15E77XXX7X70 {5, 11, 17, 37, 61, 131, 141, 841} k = = 1 mod 2 (2)k = = 2 mod 3 (3) (not full tested) 3E7320, 659XE0, 781E9X, 82E8E6, 9E2004, 13229EX, 142080X, 1534E40, 1568920, 15XX440, 16E9956, 1728496, 17400E0, 223E154, 247193X, 2517020, 28X9794, 2X51574, 31119XX, 3138950, 3254380, 343E420, 3462316, 348EX74, 34E7820, 35420X0, 35471EX, 35X6886, 35E635X, 35E8680, 35EX126, 363124X, 36446X4, 3647050, 364904X, 3658400, 3677914, 3779864, 377X94X, 37X4626, 382463X, 384374X, 3886020, 3892880, 38E52E0, 3907760, 3949956, 3953566, 3970754, 39871X4, 39E1X94, 3X00476, 3X0X08X, 3X9022X, 3E0946X, 3E2931X, 4004304, 40053E6, 4013E74, 4030386, 40539XX, 406X5XX, 4084320, 4111000, 41X347X, 420X140, 4220366, 4220EEX, 42265E6, 4226944, 4292480, 4314926, 4323054, 4335934, 43904X4, 4404X1X, 44124EX, 441X3EX, 445980X, 44E3690, 4524536, 454080X, 455797X, 455X246, 4586E06, 459183X, 4596890, 45E3900, 461X72X, 462X9EX, 468556X, 4781X66, 4798500, 47X572X, 484711X, 4877544, 48993E6, 48E3984, 49260E6, 4933316, 4961450, 4962006, 4X34406, 4X34X14, 4X54200, 4X64214, 4X8XX2X, 4E23X70, 4E308XX, 4EE230X, 50233XX, 5029220, 504050X, 506524X, 50EX910, 5115446, 5176EX6, 521EE5X, 5222644, 5234E20, 5267016, 52X353X, 5333840, 535E446, 5375424, 5391584, 53E2260, 5406006, 5433E54, 5482014, 5491E46, 55016E4, 5516X9X, 553X636, 5620E4X, 566X200, 56795X6, 5690864, 56X6894, 5739104, 5761736, 5824704, 5840906, 58X9E3X, 5902X9X, 595X946, 5989504, 5990666, 5X50610, 5X77E30, 5XE01X4, 5E07386, 5E26824, 5E30146, 5E407X4, 5E5281X, 5E698E4, 5E8877X, 601524X, 604815X, 606963X, 6097296, 61018X6, 6134064, 617650X, 618418X, 6189390, 61E41E6, 624X366, 62790E4, 6303E74, 6346884, 6348380, 637687X, 6386540, 6391780, 6396E24, 63XEE86, 63E2500, 6404334, 6417256, 6478530, 6485364, 64X5756, 6521X74, 654X164, 6557626, 6622690, 6628606, 6674620, 6712990, 671E884, 674X114, 6753096, 67624E6, 6770140, 6774720, 6783184, 68052XX, 6876116, 6879654, 6885554, 68E128X, 69013XX, 6904470, 6917464, 6983574, 69E9150, 6X2001X, 6X26710, 6X98716, 6XE2814, 6E229E4, 6E3336X, 6E35970, 6E37266, 6E4X700, 6E80254, 70067X4, 7029860, 7033X96, 7040E36, 721EE4X, 7241E7X, 7288276, 72E5X74, 735E510, 737XX24, 738862X, 7395X56, 73X7696, 73E4446, 7406430, 74X821X, 74XX174, 752X474, 753E126, 754996X, 7572450, 7578594, 7592E54, 7598666, 7610880, 7630540, 768X176, 7717966, 773391X, 7734E86, 7751360, 7775326, 7792500, 781X844, 7831E20, 7868066, 7928410, 7968526, 7X14324, 7X36874, 7X65E4X, 7E13EXX, 7E5066X, 7E5498X, 7E5EE60, 7E89990, 8085006, 8099556, 80XE030, 811690X, 81X5604, 8240E46, 8265X7X, 8268E40, 826E080, 8294660, 8295006, 83031X0, 83036X6, 8358E16, 8376X90, 839355X, 842271X, 8436724, 8440550, 8475174, 8479X16, 8514364, 8521616, 8522084, 85X8E24, 863790X, 864155X, 865607X, 86753X6, 86X2536, 8700314, 8701XX0, 87EX300, 88218X4, 8848034, 89955EX, 8X034XX, 8X14074, 8X51190, 8X938X4, 8XEX136, 8E1EX2X, 8E8232X, 8E98956, 8E99574, 8EX03E4, 903258X, 903EX1X, 9043544, 9057346, 907E370, 9081946, 90824X0, 911E1E4, 911E410, 914E896, 9160E16, 91XX1X0, 9225394, 9241E6X, 9271E44, 9274XEX, 92E46X0, 92E506X, 9372430, 9376EE6, 9391970, 9413E90, 9443X54, 944E7X4, 9455454, 9459106, 9532306, 9541410, 954E600, 9552X84, 955621X, 9626434, 9649016, 964X134, 96E10E0, 972382X, 9733980, 975109X, 9757514, 9771960, 9773274, 9773666, 97791X0, 977E214, 9819EX4, 9844836, 995672X, 9971E56, 9988910, 9997364, 99X2E36, 99X7694, 99E4686, 9X5X220, 9X6977X, 9X775X4, 9X9327X, 9XX0816, 9XX5804, 9XEE98X, 9E25086, 9E2E4XX, 9E3513X, 9EE1254, 9EE8090, X012046, X0205EX, X045E46, X05023X, X0685E6, X0X5396, X101040, X155E0X, X159E20, X1874X0, X1XX254, X2856X0, X3X6E34, X3E7130, X3EE264, X421E5X, X432790, X4661E0, X4X1986, X5X3586, X5E8240, X613296, X678860, X69E93X, X724726, X77E4XX, X7X2670, X7XE080, X819126, X851684, X863534, X864830, X872420, X8785X0, X8E827X, X920986, X933480, X9X6834, XX00X94, XX16574, XX59E50, XE82406, XE95XE0, XE98820, XEX0034, XEX074X, XEX87EX, E06E070, E070E3X, E091066, E09X986, E157130, E166306, E172X86, E21665X, E23197X, E240076, E247780, E3326X6, E34464X, E34E7X6, E360160, E382300, E39642X, E3X7770, E468X34, E4E7E34, E501966, E5112EX, E5241X6, E53E676, E621E56, E69X360, E70X976, E763626, E767666, E7X285X, E7E5570, E7E756X, E806744, E834346, E85025X, E86X934, E929680, E935300, EX17296, EX91590, EX9341X, EE0X0E6, EE2X09X, EE36650, EE87726, EE9E20X, EEEX236, ..., 7X1X20×7121920+119XE344×7114432+11846756×71076X9+11546E04×7106X61+1157147X×7E8311+16X6626×7E7292+1 8 3E {3, 5, 11} All k = m^3 for all n; factors to:(m*2^n + 1) *(m^2*4^n - m*2^n + 1) k = = 6 mod 7 (7) 0 (proven) none (proven) 27×818+13X×84+134×84+131×84+124×84+114×84+1[1] k = 1 and 8 proven composite by full algebraic factors. 9 1434 {5, 7, 11, 61} k = = 1 mod 2 (2) 1 1218 109X×931X00+11064×921633+11110×99391+190X×91E52+1642×91071+11016×9920+1 X 5387 {7, E, 11, 31} k = = 2 mod 3 (3) 1 452X 2XE0×X40726+14350×X21E36+148XX×X10289+12431×X6EEE+14629×X69X9+136X4×X6270+1 k = 84 and 6E4 are GFN with no known prime. E X42 {3, 7, 17, 31} k = = 1 mod 2 (2)k = = 4 mod 5 (5) 0 (proven) none (proven) 67X×E125E14+1X24×E13242+12X8×E7459+1732×E1X29+19X4×E1598+1442×E6X7+1 10 375 {5, 11, 25} k = = X mod E (E) 0 (proven) none (proven) 298×102X5626+1276×101470+1199×10458+129E×10267+1256×10203+131×10147+1 k = 10 and 100 are GFN with no known prime. 11 E0 {5, 7, 15} k = = 1 mod 2 (2)k = = 2 mod 3 (3) 0 (proven) none (proven) 40×113763+1X0×11X94+18X×1148+154×1122+194×1110+19X×11E+1 12 4 {3, 5} k = = 10 mod 11 (11) 0 (proven) none (proven) 1×122+1[2]3×121+12×121+1 13 X292X07695290 {11, 15, 95, 157, 181, X41, 22E1} k = = 1 mod 2 (2)k = = 6 mod 7 (7) (not full tested) X4808, 1854E6, 290X70, 794744, E46828, 1056556, 1328876, 1848708, 1858798, 1942008, 195E776, 1977000, 197EE38, 19X2962, 1X727X6, 1X729EX, 1E97990, 202537X, 2032680, 2052134, 2069870, 20E0958, 215X10X, 21X5966, 2211138, 2306784, 2311068, 2575690, 264E578, 27222E4, 276E740, 28080X4, 2832618, 2962668, 296685X, 2978400, 2997482, 29E3XE4, 2X18338, 2X64468, 2E05588, 3046484, 30X529X, 30E2152, 314X2E8, 3170574, 3251X72, 3252E6X, 330917X, 343X258, 3445446, 345E108, 34755E8, 34X5756, 3520560, 361627X, 36197XX, 3625X24, 3630X70, 3692512, 37E27E0, 385E642, 3875616, 3913X18, 397301X, 4000510, 4011EX2, 413X3XX, 4141886, 4152822, 42285E8, 4283202, 4325590, 4381916, 43E3X40, 4499366, 4503042, 4532864, 4541366, 461X45X, 47305X0, 4745306, 4916848, 49178E6, 4920144, 4930916, 4941540, 4952258, 4957280, 4X41654, 4X57108, 4X68610, 4X9XX36, 4XX23X2, 4XE92X0, 4EEE806, 5007E72, 50E72X0, 51208X4, 5149X28, 517010X, 5196374, 5210EE8, 5263212, 531E878, 5361222, 5427X34, 54383X0, 545XEEX, 548012X, 578X866, 5792818, 5877784, 5886970, 592060X, 5975764, 5976856, 5X12966, 6036086, 6077448, 6103568, 611X3EX, 6144924, 61768E2, 6243724, 6317412, 6326774, 6360572, 639EX38, 63X9110, 6434202, 6454570, 6455X3X, 64E7940, 64E8082, 6517556, 6568190, 664E490, 67021E8, 6889EE4, 68XE920, 68E0E16, 6967376, 69E349X, 6X3X338, 6X48968, 6X4E834, 6E77514, 6E87102, 6E9X882, 7011788, 702466X, 705278X, 70E2424, 7108184, 7279196, 72E6XEX, 7370156, 7380E66, 748E326, 7524326, 7620EE2, 771777X, 776X790, 777E416, 7790X02, 7816232, 78EX244, 7958640, 7X261E2, 7XE3924, 7E58E0X, 7EX5678, 8026X24, 803X1X0, 8127980, 8153840, 8195E98, 8369004, 8413930, 847E966, 85E8E1X, 862X924, 8685264, 8798116, 8850XX2, 8910446, 8916632, 896X476, 8X048X6, 8E65798, 8E794X2, 8E93156, 9015EXX, 901X2E2, 90E1114, 912X406, 9163508, 9276358, 930E6EX, 9346772, 9359842, 939EX82, 93E6768, 9411850, 9459172, 94711X8, 9479X76, 9564044, 958E6E8, 9675326, 9676778, 96XEX42, 9773346, 97X6712, 9805E34, 997X728, 99X0X5X, 9X10X46, 9X18138, 9X271EX, 9X28482, 9X9EXE8, 9XX0E60, 9E43530, 9E4XE7X, 9E50892, 9E5E798, 9E8407X, X016010, X230014, X288X06, X2X4840, X40286X, X4E8X62, X502698, X625582, X68E822, X803038, X931510, XX8148X, XXXX49X, E02X322, E056E58, E068604, E0E4528, E30XEX6, E348814, E37001X, E37025X, E4206X2, E42X5E8, E45X994, E477802, E4X2602, E500068, E565E22, E5820X4, E587856, E61840X, E681312, E743924, E777956, E83E6E4, E9X7216, E9E2670, EX65898, EX760XX, EE9E404, EEE2040, ... 1361364×139520E+176171X×139013X+12E4E10×13885E1+116E39X0×134070X+11194764×1339783+11481556×1335933+1 14 32759 {7, 11, 15, 181} All k=4*q^4 for all n: let k=4*q^4 and let m=q*2^n; factors to: (2*m^2 + 2m + 1) * (2*m^2 - 2m + 1) k = = 2 mod 3 (3)k = = 4 mod 5 (5) 12 1824, 3X33, 5X87, 9XX6, 10311, 12286, 15711, 1888X, 1E476, 208X1, 21597, 295X3, 2X91X, 31490 E9EX×142739X41+12800X×14926382+128XXX×147X4X66+117591×14709300+1104X4×1450E321+1116X3×14390EEE+1 k = 1544 and 1E194 proven composite by full algebraic factors.k = 31E14 is GFN with no known prime. 15 1E2 {3, 5, 25} k = = 1 mod 2 (2) 1 184 19X×1590100+1114×1580114+178×152583E+174×152998+1X×15950+111X×15750+1 16 292 {5, 11, 17} k = = 14 mod 15 (15) 0 (proven) none (proven) X2×161211EX+1279×1611E50+1203×161493+131×16321+1262×16196+1X3×16178+1 k = 16 and 230 are GFN with no known prime. 17 30X986 {5, 7, 11, X7, 541} k = = 1 mod 2 (2)k = = 2 mod 3 (3) 395 X06, 1566, 16X4, 2196, 2736, 2806, 3E24, 62X0, 8510, 8XXX, 8XE0, 9054, 949X, X224, E006, E62X, E91X, E950, 10680, 123XX, 128XX, 1526X, 15374, 15404, 15584, 19186, 1X244, 1X786, 1X890, 20X1X, 21556, 21636, 22200, 22646, 2287X, 25010, 2681X, 28164, 29374, 2X326, 2E28X, 2E740, 30X9X, 3134X, 314EX, 31526, 31710, 32464, 3297X, 33520, 33EE6, 35X80, 36146, 36880, 38206, 39004, 3X08X, 3X960, 3X9XX, 3E3E6, 4034X, 40874, 40930, 40970, 41720, 4203X, 42176, 42976, 42E0X, 44114, 44120, 45334, 47080, 48080, 4887X, 49444, 4EX94, 500X6, 502X6, 5045X, 51654, 523X6, 53140, 5406X, 54254, 5611X, 56330, 5805X, 58760, 59560, 597X6, 59894, 5X86X, 5XE00, 5EE36, 618XX, 61904, 6337X, 63434, 63E50, 6424X, 643X6, 64830, 64EE0, 678XX, 6X050, 6X630, 6X680, 6X75X, 6E34X, 6E560, 71504, 72476, 73620, 758EX, 77290, 7821X, 782E0, 794E4, 79506, 7X296, 7X414, 805E6, 82526, 83464, 8364X, 8385X, 8436X, 856EX, 86154, 8663X, 866X4, 87914, 88190, 900EX, 91594, 93436, 93694, 93790, 95690, 96336, 97684, 99106, 9939X, 9XE80, 9E5EX, X3190, X3576, X4344, X4844, X77E0, X8260, X9326, XX186, XE074, XE45X, E012X, E0516, E0866, E0X90, E168X, E2004, E2X2X, E5830, E5936, E6560, E6850, E75X4, E86EX, E9E20, E9E36, EX286, EX324, EE22X, 10014X, 100264, 100310, 100504, 100EXX, 101100, 1011E0, 101346, 101810, 101926, 101940, 102X24, 103306, 103866, 104286, 105344, 105650, 107276, 107436, 10800X, 108164, 108550, 10E634, 10EE5X, 111664, 111786, 113134, 113X20, 113E50, 114110, 114150, 11432X, 115E04, 117906, 11823X, 11E034, 11E340, 120E3X, 122756, 122X2X, 123290, 12332X, 123814, 123830, 12548X, 1289EX, 12X68X, 12E5EX, 12E826, 12E850, 130X56, 130X8X, 131984, 131X9X, 133026, 133980, 134054, 134150, 134X1X, 13544X, 135836, 136040, 138674, 139190, 13XX30, 1402XX, 141324, 14171X, 14409X, 14584X, 145880, 145X84, 146116, 146E8X, 14702X, 14844X, 14E26X, 14E614, 150586, 150X2X, 152146, 152X66, 153084, 1548E6, 156134, 157636, 157646, 15813X, 158330, 1595EX, 1596X6, 159956, 15X174, 15X3EX, 15E846, 1602XX, 1613E6, 161726, 1622EX, 16234X, 164626, 165024, 1661X6, 166404, 166590, 166910, 169896, 16E9EX, 17165X, 171XX4, 171E84, 172310, 172716, 173226, 173266, 1752EX, 175326, 175EE6, 1769E6, 17X084, 17X594, 17E044, 17E73X, 181E60, 182200, 182486, 1832E0, 183E80, 184094, 1881E6, 188930, 18E864, 190E7X, 191906, 191XEX, 191EE4, 193424, 193X30, 1941E0, 1941EX, 194E70, 195X74, 196140, 198144, 19829X, 19865X, 1990E4, 19E210, 19E22X, 19E866, 1X05E0, 1X0690, 1X32X0, 1X4656, 1X4770, 1X5166, 1X725X, 1X7516, 1X921X, 1X972X, 1X9X6X, 1XX98X, 1XE95X, 1E0080, 1E0356, 1E0580, 1E0E2X, 1E1540, 1E221X, 1E3714, 1E457X, 1E5790, 1E57XX, 1E6614, 1E6776, 1E8206, 1E8300, 1E848X, 1EX590, 1EE04X, 1EE294, 1EE654, 201394, 201690, 202206, 2051E4, 20598X, 20602X, 206E0X, 209566, 2098E6, 20X4X6, 211006, 2115X6, 212X5X, 213326, 21566X, 218EE6, 21906X, 21X080, 21X274, 21XE6X, 2208E6, 222674, 223984, 223X8X, 227384, 2280E4, 229346, 229E96, 22E9E4, 230490, 231056, 23199X, 232644, 232XE6, 2330XX, 233164, 234010, 234220, 234236, 234796, 234X6X, 235344, 236214, 236476, 236596, 236780, 23679X, 237104, 237410, 237E54, 239114, 239710, 24226X, 24356X, 244634, 244E76, 244E94, 245226, 24583X, 245896, 246034, 246710, 247140, 247EE6, 248764, 2508XX, 251650, 252096, 252240, 252E84, 25309X, 2544X4, 254904, 254E06, 2552E0, 25539X, 2555X0, 255884, 255930, 255980, 256300, 25630X, 257184, 257476, 258820, 25E6E6, 25E730, 261276, 26146X, 261564, 261E54, 26218X, 262244, 264464, 265070, 265626, 2657X6, 2690EX, 26E196, 26E284, 26E646, 26E696, 27151X, 27228X, 273634, 274284, 2745X6, 276690, 27873X, 278766, 278EXX, 279104, 279880, 27996X, 27X180, 27X840, 27XX34, 27E420, 27E9E4, 28044X, 281110, 282370, 28320X, 283500, 283816, 283X40, 28578X, 286224, 287XE4, 28826X, 288946, 2893E6, 28X650, 28X6XX, 28X704, 28E99X, 290314, 292736, 292X6X, 29381X, 2950E6, 296016, 2976X6, 297964, 298754, 299216, 299556, 29X120, 29E0E6, 2X047X, 2X0X60, 2X12X6, 2X1956, 2X3806, 2X3976, 2X4700, 2X677X, 2X7434, 2X950X, 2XX114, 2XE86X, 2XEX10, 2E0926, 2E14EX, 2E1X4X, 2E2196, 2E2794, 2E4E90, 2E54X0, 2E633X, 2E6E20, 2E843X, 2E8964, 2E9206, 2EX5E4, 30009X, 30067X, 3006X6, 30386X, 304934, 304994, 305180, 305584, 306930, 307094, 30736X, 3076X0, 309136, 30X84X, 30X92X 29911X×178X542+1211854×178X017+1216530×1789739+1149200×1789306+12E755X×1788E58+1178X40×17883E2+1 18 8 {3, 7} k = = 16 mod 17 (17) 0 (proven) none (proven) 6×1813+17×182+14×182+11×182+1[3]5×181+13×181+1 19 6E6 {E, 11, 15} k = = 1 mod 2 (2)k = = 4 mod 5 (5) 0 (proven) none (proven) 9X×19E5X1+164X×19172+1514×1915E+1694×1970+1696×1937+1406×1933+1 1X 3X5X {5, 1E, 81} k = = 2 mod 3 (3)k = = 6 mod 7 (7) 1 2E74 E23×1X2E77X4+11130×1X151755+12549×1X127E52+13337×1X486X6+138X6×1X22403+12E19×1X11E00+1 k = 1X and 344 are GFN with no known prime. 1E 132 {3, 5, 45} k = = 1 mod 2 (2)k = = X mod E (E) 0 (proven) none (proven) 58×1E157447+18×1E58EX7+1X2×1E8169+1X4×1E197X+110X×1E1816+168×1E3EE+1 20 158X3 {5, 7, 11, 61, 67} k = = 1X mod 1E (1E) 51 468, 777, 10X3, 10E4, 1304, 143E, 15E6, 1E78, 205X, 2109, 27EX, 29EX, 2E75, 30E0, 30E8, 314X, 34X9, 355X, 4264, 4596, 4658, 47E2, 4823, 4881, 5453, 5474, 5745, 5761, 5976, 5X64, 6117, 6696, 7609, 7627, 7E30, 9269, X046, X38E, X418, X506, X650, XE3E, E01E, E072, E84E, EEE7, 10503, 10X95, 10E30, 11013, 11970, 11996, 12414, 13038, 1325E, 139X8, 14X03, 14E99, 15074, 1513X, 15169 8114×20171X8E+12286×20165E46+111X65×20154452+14X15×20153E77+1218X×20150984+174X7×201503E7+1

## Riesel problem base b

### Table for bases up to 20

All n must be ≥ 1.

k-values with at least one of the following conditions are excluded from the conjectures:

1. All n-values have a single trivial factor.

2. Make a full covering set with all or partial algebraic factors.

k-values that are a multiple of base (b) and where k−1 is composite are included in the conjectures but excluded from testing. Such k-values will have the same prime as k / b.

 base conjectured smallest Riesel number covering set ks that make a full covering set with all or partial algebraic factors trivial ks number of remaining k remaining k top 6 primes comments 2 206817 {3, 5, 7, 11, 15, 181} (none) 43 13E1, 5405, 11845, 1652E, 1X321, 23007, 32X11, 3728E, 3XX95, 4637E, 4826E, 52157, 5X655, 627X7, 6X947, 70995, 79685, 9380E, 9E29E, 9E41E, X4817, XXX61, E3747, EX135, 100837, 1329E5, 134X9E, 13760E, 138197, 13975E, 142E1E, 1464X7, 14761E, 14E252, 155X15, 156327, 157207, 15739E, 15924E, 15E271, 16664E, 167X15, 171917, 179161, 195391, 1X8111, 1XX70E, 1E0467, 1E08E2, 1E4EE1, 1EX387 112555×22EX9280−1202X11×224X42XE−1174E4E×2249E080−11E5E1×22344139−1128067×222847X5−1172407×221961X3−1 3 1028027E722 {5, 7, 11, 15, 17, 31, 35, 141, 531} k = = 1 mod 2 (2) 73008 129449X, 2378918, 3607022, 36E7288, 3743E2X, 5799174, 60X83X8, 71X60E2, 9913X78, X4X12X2, 10X3718X, 115557X4, 11641842, 1537XX92, 15532614, 16X0497X, 17950744, 17X1699X, 1898066X, 18X9447X, 19X6E322, 1EX3E814, 2561E3XX, 27673X84, 28839858, 2X64145X, 2X808334, 300E6E12, 31229342, 3267E738, 344567X8, 3493X544, 3562727X, 365675X4, 36E05E42, 38802584, 388X0244, 3964849X, 3X5794E8, 45623128, 45720874, 47733170, 4EE18XE2, 5049601X, 52850E42, 5618103X, 5806E38X, 58998584, 5X22EE42, 5E02X43X, 5E820694, 62498228, 633040E4, 64301548, 65951X5X, 66465312, 67X04768, 68215612, 68360328, 6886X732, 696E8E14, 69951564, 69E9X428, 6X38E168, 6E235E0X, 701X8282, 710486XX, 719X6362, 71E37988, 762EX038, 76494672, 7697E61X, 76XEEXE8, 783E3X4X, 78E29104, 7931E782, 80728XX4, 80926704, 841346E8, 855320E8, 8654E44X, 8740X544, 8761X204, 88002982, 8X0441X8, 8X77E302, 91457178, 91X4985X, 944E5862, 94853X30, 9485571X, 96503274, 96910634, 9855745X, 9X47286X, X0573678, X072942X, X130X44X, X132X9X4, X19170X4, X22005E4, X2639998, X30E803X, X9E6067X, XX3091X4, XX93239X, XE8X09X2, E5904524, E7526034, EX599EE2, ..., 1027421297X, 102746E7262, 10275802472, 1027608295X, 1027620E62X, 10276395E2X, 10276679634, 1027879E248, 1027899X56X, 1027E3X3762, 1027E936644, 1027E99575X 63005504×334E917−1359134X34×334E809−11XE26EE14×334E318−110163226X×334976X−12E8X6222X×3349316−1173838204×3347XX0−1 4 1E143 {5, 7, 11, 17, 61, 91} All k = m^2 for all n; factors to:(m*2^n - 1) *(m*2^n + 1) k = = 1 mod 3 (3) 7 27X2, 5405, 5613, 8389, E320, 11845, 1652E E289×46250X1−1E729×4348262−146E×417376X−18026×415E164−19826×476930−11X885×46E084−1 k = 3^2, 6^2, 9^2, 10^2 (etc. repeating every 3m) proven composite by full algebraic factors. 5 148842 {3, 7, 11, 27, 421} k = = 1 mod 2 (2) 53 211X, 2X0X, 11X02, 13212, 18494, 26762, 30E3X, 31472, 33518, 3520X, 3822X, 3XE52, 45948, 4E602, 4E6E4, 53272, 53692, 536E2, 6171X, 64374, 6584X, 67204, 6E154, 7021X, 70238, 70E18, 71684, 734XX, 74264, 75738, 782X4, 7E498, 8272X, 82XX4, 86X3X, 8967X, 908E8, 9199X, 92192, 95428, 9892X, 9X34X, 9E892, X3X04, E20X2, E641X, E6612, EX5EX, EEX02, 105E14, 10991X, 10X82X, 110E0X, 112452, 11933X, 119428, 124862, 127E18, 13125X, 134232, 138742, 142132, 14412X X00E2×510162X6−1E6172×5EE8252−170788×5EX50XX−118888×5EX0552−113676X×5E30X17−143224×5E2EE45−1 6 41013 {7, 11, 27, 31, 81} k = = 1 mod 5 (5) 1 E11 19344×66E1333−121562×62356X2−138EX7×6230609−125635×621602E−128EEE×61E3X09−13983E×61X1142−1 7 670E5X1X7X2 {5, 11, 17, 37, 61, 131, 141, 841} k = = 1 mod 2 (2)k = = 1 mod 3 (3) (not full tested) 1328X0, 554896, 661746, 851430, X03428, X25X22, X6XE60, 1191180, 12234E6, 1250398, 1374500, 1545782, 1547X40, 1565162, 1574548, 1676506, 1726256, 176X746, 1808230, 186X842, 19523E2, 1X27X48, 1X83496, 2129882, 21E8218, 2237486, 2262148, 2430520, 2471X16, 2550566, 25E1548, 2643362, 2846E86, 296E072, 2987030, 2X88150, 2E12418, 2E147X6, 30279X6, 3328998, 3360648, 3380598, 346X978, 34701E8, 351EXX8, 3542466, 35935E6, 3621462, 3809E82, 3926X08, 39681E6, 3985186, 3X38392, 3XE6162, 3EEEE16, 4005220, 401XE82, 4036796, 405X548, 40E8218, 4138700, 4149328, 4354632, 4501638, 4538420, 4564EE8, 4696942, 4820262, 4863X50, 49E0400, 4XXE626, 4E66622, 4E9X972, 50X0460, 5170890, 52X71E2, 5321022, 5357156, 54E6418, 5574970, 55E1X08, 5784E80, 5794E28, 5816928, 5987940, 5E368E2, 5E74752, 6102150, 6174696, 6315876, 635E862, 6443XE0, 6574918, 6655158, 675EE28, 67EX958, 6X60120, 6E72E92, 7045748, 7083506, 7094536, 70X7710, 7216X40, 7480192, 7616XX0, 765X5E0, 77224X0, 77371X2, 7740842, 7771858, 7821E60, 7X9X992, 7E86572, 800XE92, 802XX16, 80XX352, 815X220, 8245528, 82X4362, 82E8056, 8504230, 8634150, 8654388, 8702E16, 8704X48, 8717992, 897E3X2, 8X960X2, 91E37X8, 92EE210, 9362898, 938E482, 9401X28, 96315X2, 96X1688, 9707246, 97115E0, 9771386, 9881400, 9903E02, 9977026, 9E60298, 9E71292, X236538, X2E88X2, X354410, X379952, X38X690, X666576, X6817E8, X823270, X8403E0, X8950X6, XXE7552, XE25540, E1305E8, E169398, E195756, E224X46, E382090, E437346, E44E932, E6613E2, E669620, E69X162, E794410, EX1E320, EX765X2, EX83488, EXX7586, EXX9156, EE22286, EEX5616, ... 139E42×7124717−1260828×7E6E10−155X4X0×7E2392−1345666×792322−19915166×772415−110662888×772400−1 8 12 {3, 5, 11} k = = 1 mod 7 (7) 0 (proven) none (proven) E×816−15×84−110×83−17×83−12×82−111×81−1 9 62 {5, 7, 11, 61} All k = m^2 for all n; factors to:(m*3^n - 1) *(m*3^n + 1) k = = 1 mod 2 (2) 0 (proven) none (proven) 20×98−112×98−150×95−136×95−138×94−13X×93−1 k = 4, 14, 30, and 54 proven composite by full algebraic factors. X 5X80 {7, E, 11, 31} k = = 1 mod 3 (3) 1 2685 408E×X366025−14E6E×X160010−13X35×X2XX48−11153×X25EE8−11063×X22676−193E×X1523E−1 E 5EX {3, 7, 17, 31} k = = 1 mod 2 (2)k = = 1 mod 5 (5) 0 (proven) none (proven) 52×E131E6−1218×E310−1124×E137−11E8×E136−1372×E66−1328×E66−1 10 274 {5, 11, 25} (Condition 1):All k where k = m^2and m = = 5 or 8 mod 11:for even n let k = m^2and let n = 2*q; factors to:(m*10^q - 1) *(m*10^q + 1)odd n:factor of 11(Condition 2):All k where k = 3*m^2and m = = 3 or X mod 11:even n:factor of 11for odd n let k = 3*m^2and let n=2*q-1; factors to:[m*2^n*3^q - 1] *[m*2^n*3^q + 1] k = = 1 mod E (E) 0 (proven) none (proven) 111×101E9−13X×10142−1214×1034−1197×1034−17X×1030−1204×1026−1 k = 21, 54, and 230 proven composite by condition 1.k = 23 and 210 proven composite by condition 2. 11 212 {5, 7, 15} k = = 1 mod 2 (2)k = = 1 mod 3 (3) 0 (proven) none (proven) 200×1153255−1102×1126−178×111E−186×1118−1210×11X−1160×11X−1 12 4 {3, 5} k = = 1 mod 11 (11) 0 (proven) none (proven) 2×124−13×121−1 13 40E498E248436 {11, 15, 95, 157, 181, X41, 22E1} k = = 1 mod 2 (2)k = = 1 mod 7 (7) (not full tested) 164X96, 1506E08, 1611X60, 1906942, 251E298, 2X30E62, 3071432, 3326098, 3457132, 35294X4, 3532268, 353XX60, 36187X2, 361XE52, 3665894, 36XE112, 372E10X, 3779834, 388241X, 38XE852, 3980114, 3989556, 3X67E6X, 3X7099X, 3XXE808, 3E62E18, 3E9805X, 3EXEEE2, 3EE54E2, 4029152, 4101770, 410213X, 411078X, 411X984, 41454E8, 4212502, 42687X2, 42X8556, 4308326, 4335436, 4340878, 4352722, 4432X26, 4513516, 452325X, 45E5X70, 490242X, 491733X, 4942698, 4970446, 4X25EX8, 4E4954X, 4E4X088, 500001X, 5102206, 514E280, 51X4764, 5226226, 52650X8, 5309008, 5331E04, 53E5E2X, 5449410, 54XE06X, 552E994, 557E3X6, 5711830, 57E8E38, 5818444, 583E778, 5969394, 5X13666, 5X17E80, 5X21874, 5X38126, 5X664X2, 5E30E68, 5E44604, 602702X, 612EE12, 614E450, 6183592, 622XX80, 6235X54, 6239358, 62775X8, 6323014, 639629X, 63E0994, 64EE640, 669E894, 6728X6X, 6827384, 6839426, ... 1E63996×13188060−1323X126×131614X3−11155348×13148X59−1137670X×131135XE−11915466×13105017−11883X06×13100545−1 14 177X5 {7, 11, 15, 181} All k = m^2 for all n; factors to:(m*4^n - 1) *(m*4^n + 1) k = = 1 mod 3 (3)k = = 1 mod 5 (5) 14 30E, 13E5, 5613, 78E0, 7E1E, X748, E320, E538, 10583, 11845, 12563, 13322, 1491E, 15365, 1652E, 1713E 734E×14258327−12118×14190042−1E729×14184131−14535×14157E47−11771E×14E49E0−13E70×14X53E7−1 k = 3^2, 10^2, 13^2, 16^2, 23^2, 26^2, (etc. pattern repeating every 26m) proven composite by full algebraic factors. 15 72 {3, 5, 25} k = = 1 mod 2 (2) 0 (proven) none (proven) 38×153908−130×15183−1X×1599−122×1592−14X×152E−13X×1521−1 16 186 {5, 11, 17} k = = 1 mod 15 (15) 0 (proven) none (proven) 107×162XX−166×16124−142×1692−167×1653−1179×1638−1134×1638−1 17 4600X2 {5, 7, 11, X7, 131} (Condition 1):All k where k = m^2 and m = = 2 or 3 mod 5:for even n let k = m^2 and let n = 2*q; factors to:(m*17^q - 1) * (m*17^q + 1)odd n:factor of 5(Condition 2):All k where k = 17*m^2 and m = = 2 or 3 mod 5:[Reverse condition 1] k = = 1 mod 2 (2)k = = 1 mod 3 (3) 670 266, 2332, 4772, 6X80, 7996, 9780, 9E18, X418, 101X6, 102E8, 10E30, 11518, 12086, 12088, 12168, 13E60, 14812, 15760, 15E46, 17906, 17X86, 1E438, 1E480, 20730, 20818, 20X02, 21416, 22848, 23356, 24922, 24996, 249E2, 25012, 256X8, 25E66, 26900, 270X6, 28068, 28452, 29738, 2E408, 2EX30, 30120, 30646, 31130, 313X8, 325X0, 33168, 33826, 35558, 35898, 36242, 364E2, 36688, 36786, 36898, 37122, 37316, 387X6, 39432, 3X726, 3E9E0, 40858, 40E98, 41020, 42048, 42522, 42756, 43152, 43250, 44398, 45892, 46922, 47602, 47758, 4X6X8, 4XE98, 4E212, 4E440, 51170, 51338, 51820, 51936, 52146, 52XE6, 53882, 54262, 54596, 55E42, 56292, 562E8, 56556, 565X8, 56620, 56E70, 58XX2, 59560, 5X232, 5E196, 5E8X6, 61148, 621X8, 62482, 63126, 641E2, 64418, 65202, 66130, 66288, 67336, 6X688, 6X702, 6XE56, 6E318, 71520, 72008, 73XX0, 74448, 74672, 75302, 75836, 767X6, 76978, 781E2, 78832, 79260, 79502, 7X096, 7X192, 7X200, 80E36, 81E22, 82X42, 83088, 838X2, 86550, 86698, 87550, 87858, 88046, 89038, 8E402, 8E526, 904E2, 91192, 92802, 93152, 934E8, 93638, 936X2, 93EE0, 94202, 95712, 96828, 97298, 974E8, 97556, 99816, 9X412, 9X5E0, 9E060, 9E432, 9E722, 9E802, X1250, X1888, X2750, X3332, X4352, X5080, X5312, X7306, X7308, X7500, X75X0, X7638, X9126, X9X36, XE558, E0188, E3642, E4240, E5890, E67E2, E68E8, E7490, E8810, E8878, EX092, EX8X6, EXE72, EE808, 100510, 101128, 1012E6, 101426, 1017X6, 101XE2, 103138, 103452, 103910, 103X16, 104E56, 105140, 105X20, 105E90, 106298, 106572, 1065X6, 1070X8, 107768, 108410, 108880, 1090X0, 109550, 10X770, 10E492, 10E926, 110478, 110486, 110X32, 111928, 112488, 112X52, 115250, 115EE0, 116806, 116928, 1171X6, 1175E6, 117E48, 118192, 118X60, 118E18, 11X006, 11X0X6, 11EX58, 121X38, 121XE0, 122E52, 123066, 124718, 126140, 126676, 1276E8, 129170, 129710, 129728, 129E62, 129EE2, 12X148, 12X6E6, 12E908, 12EE90, 1302X8, 130418, 130812, 131170, 131E52, 132366, 132500, 132XX0, 133190, 133228, 1335E0, 134720, 135208, 1356E8, 135X98, 136218, 137068, 137XX8, 138376, 138542, 138702, 141068, 141808, 142100, 1423E0, 142906, 1441E2, 144226, 144232, 144478, 144968, 145008, 1451X6, 145576, 147772, 148X06, 14XXX0, 14XE80, 151178, 151878, 151950, 154736, 154818, 156178, 156X22, 158138, 1582E0, 158X68, 15E070, 15E708, 15EE02, 160218, 160766, 161646, 161808, 162472, 162488, 162536, 163948, 164878, 164E26, 166460, 168932, 16X742, 16X992, 170060, 170560, 170858, 170E98, 171328, 172442, 172852, 173708, 173EE2, 174000, 175148, 175358, 176192, 1771X0, 177440, 177652, 177758, 178508, 1787X0, 1788X6, 1794E0, 179858, 180222, 180598, 180860, 181910, 182648, 182898, 1833E6, 183760, 183E80, 185022, 186602, 187576, 187906, 188EX8, 18XX30, 18XX46, 18XE00, 18E726, 190106, 191866, 192432, 192666, 1943X6, 194572, 194700, 195042, 195648, 195728, 195E46, 196116, 196158, 196390, 196542, 197722, 1988E0, 198966, 198E60, 199152, 199XX8, 19X016, 19X568, 19E172, 19E710, 1X0118, 1X0682, 1X0692, 1X3058, 1X30E6, 1X4488, 1X4956, 1X50E2, 1X5860, 1X5876, 1X5EX8, 1X6186, 1X6570, 1X6E18, 1X7102, 1X7450, 1X8026, 1X8630, 1X9690, 1X9788, 1XX140, 1XEEE0, 1E1946, 1E3386, 1E3836, 1E4110, 1E4716, 1E5530, 1E5X48, 1E6988, 1E8060, 1E8212, 1E8690, 1E8852, 1E8X46, 1E9506, 201X12, 202730, 203870, 2038E2, 2043E6, 204566, 205536, 205728, 207X06, 207E68, 209120, 2098X8, 20XEX8, 20E770, 20E852, 210716, 211292, 211348, 211768, 212122, 2132X2, 213930, 213E16, 214480, 214X20, 214E90, 215298, 215802, 216396, 2179E2, 217E88, 219392, 21X0X6, 21X222, 21X932, 21X976, 21E196, 2202E0, 2212E0, 222E66, 222EE6, 2249E0, 226438, 226E08, 226E16, 226E32, 227466, 227E42, 228E52, 229170, 22XX92, 22E092, 230886, 231X30, 232346, 232662, 233236, 234552, 235220, 235272, 235416, 236968, 236E50, 237258, 237372, 237766, 238470, 239390, 239E56, 23XX02, 23EEX0, 240320, 240322, 240960, 241486, 241X18, 242368, 242E58, 244780, 244976, 2450X6, 246382, 247560, 248108, 248720, 24E910, 250018, 251078, 251406, 2521E8, 252440, 252618, 253136, 2538E2, 2539E0, 255E80, 256966, 257352, 258428, 258468, 2588X6, 259162, 25E246, 25E818, 25E972, 260676, 260X20, 2617E8, 261950, 264020, 264756, 264860, 264922, 265312, 265838, 265966, 2664E8, 266830, 267170, 2678E6, 268XX0, 26X360, 26X602, 26XE18, 26E5E6, 2708X8, 271172, 271358, 271578, 271962, 271E58, 272552, 272802, 273916, 277088, 2772E2, 277430, 277546, 277836, 277932, 278062, 279102, 279678, 279898, 279992, 27X486, 27XX08, 27XEX6, 27E720, 27EX02, 280270, 281838, 281940, 282700, 283778, 284190, 285140, 286096, 286766, 286X06, 287822, 28X052, 28E458, 28EE92, 291E46, 293182, 293406, 293560, 293562, 294780, 294992, 295058, 296236, 298640, 29X040, 29X098, 29E612, 29EXE6, 2X04E2, 2X3200, 2X5246, 2X5858, 2X63E0, 2X6448, 2X65E0, 2X8688, 2X97X2, 2X99X2, 2XX270, 2XX288, 2XX682, 2XE078, 2XE5E2, 2E0328, 2E0378, 2E0932, 2E1080, 2E1556, 2E1718, 2E25X8, 2E3826, 2E3X26, 2E4756, 2E6016, 2E6398, 2E65E6, 2E69E0, 2E7202, 2E7750, 2E7X92, 2E8180, 2E8868, 2E8980, 2E9608, 2E9930, 2E9X38, 2EE992, 300622, 301510, 302002, 303232, 303692, 303E26, 3059E0, 306210, 306866, 307EX0, 30X058, 30XE22, 30E738, 312322, 312768, 3131X6, 315070, 315762, 315XE2, 316X48, 317136, 317862, 317XX6, 319368, 3193X8, 31X360, 31E006, 31E150, 31EX30, 320008, 320056, 320362, 321526, 322156, 3225X0, 3225X2, 323X36, 324X22, 324X38, 3254X0, 326172, 326356, 327132, 3273X0, 327622, 328XE0, 329442, 32X506, 32X902, 32E0X8, 32E778, 32E786, 32E830, 330X86, 3321X0, 332432, 333396, 333X90, 334E96, 335866, 336756, 338072, 338718, 33E376, 343042, 3433E8, 343X36, 344578, 3453E0, 346812, 346E02, 347252, 347988, 348300, 348762, 348E48, 349296, 349826, 349876, 34X142, 34X618, 34XX62, 34E216, 34E928, 34EX66, 34EE38, 350196, 350526, 351340, 3513X0, 352052, 3525E2, 353X30, 356092, 357342, 357498, 358048, 358126, 3592E6, 35X566, 360666, 360XX2, 361E60, 3638X6, 364326, 3643E6, 3653E0, 365432, 365730, 366230, 366X98, 369560, 36X038, 36E730, 36EE36, 371226, 371X58, 371E56, 372140, 372972, 374088, 3769E2, 376X66, 377238, 377430, 378078, 378096, 378482, 378498, 379150, 37X402, 37X7E6, 37X950, 37EXX8, 380156, 380980, 3819X8, 382388, 383978, 384662, 385202, 385576, 385800, 385E56, 386086, 386220, 387740, 388178, 3882X0, 388610, 389372, 38X430, 38X796, 38E176, 38E9X8, 390396, 390X72, 391E28, 3923X6, 393656, 393680, 394336, 394828, 394932, 395232, 396516, 397638, 397726, 397760, 398060, 398270, 398576, 399480, 399520, 39E830, 3X0198, 3X0742, 3X0806, 3X3240, 3X3570, 3X9846, 3X9X72, 3X9E62, 3XX5E6, 3XE318, 3E0182, 3E14X0, 3E2030, 3E3170, 3E3286, 3E5230, 3E5980, 3E6838, 3E6E36, 3E7370, 3E9EX8, 400040, 400362, 401218, 401382, 402X50, 403422, 403432, 403722, 4040E2, 406670, 406960, 406E72, 407106, 407X02, 408458, 408586, 409828, 409EE0, 40X522, 40E2E8, 40E380, 410438, 410496, 4105E0, 411238, 411860, 412418, 412668, 412X62, 414E48, 4150E8, 415210, 415666, 415E22, 416X70, 416E32, 417642, 417836, 418396, 418598, 419E38, 41X7E2, 41EE16, 420012, 420386, 420660, 421008, 422E56, 422EX6, 4233X8, 4240E2, 424842, 425X88, 426516, 427202, 42X298, 42XX62, 42E0X8, 42E1E2, 4306E2, 431X10, 432238, 4323E8, 432X20, 432X60, 432EX0, 433206, 433232, 433602, 434058, 434282, 434E62, 435252, 435840, 435X16, 436068, 436752, 436X42, 437088, 4383X2, 438802, 438X06, 439478, 43E666, 4401E8, 440966, 441E50, 442790, 4430X8, 443986, 444E66, 445182, 447582, 447XE6, 448286, 44E102, 44E908, 450610, 4506E2, 450750, 451420, 4523E6, 4538X2, 4542E6, 455428, 455926, 4564X2, 457550, 458X62, 459908, 459E86, 45X1E0, 45X4X0, 45E508, 45E680 1X48×17X6026−11E02×175X3X0−1374708×174998E−12110E6×1749891−1237646×17497XE−1382136×1749711−1 k = 10^2, 16^2, 24^2, 36^2, 40^2, 4X^2 (etc. pattern repeating every 26m) proven composite by condition 1.k = 17*10^2, 17*16^2, 17*24^2, 17*36^2, 17*40^2, 17*4X^2 (etc. pattern repeating every 26m) proven composite by condition 2. 18 8 {3, 7} k = = 1 mod 17 (17) 0 (proven) none (proven) 2×18X−16×182−15×182−17×181−14×181−13×181−1 19 3X8 {E, 11, 15} k = = 1 mod 2 (2)k = = 1 mod 5 (5) 0 (proven) none (proven) 54×1917XE−1352×19696−110X×1987−170×1974−1EX×1940−1316×192X−1 1X 26E9 {5, 1E, 81} k = = 1 mod 3 (3)k = = 1 mod 7 (7) 1 2148 1968×1X79344−12395×1X19232−11799×1X14233−1705×1X13103−12472×1X718E−1135×1X6749−1 1E 338 {3, 5, 45} k = = 1 mod 2 (2)k = = 1 mod E (E) 1 298 142×1EX2230−1E2×1E141E8−128X×1EE809−1222×1E9EE0−1328×1E4450−1172×1E3730−1 20 16868 {5, 7, 11, 61, 401} (Condition 1):All k where k = m^2and m = = 2 or 3 mod 5:for even n let k = m^2and let n = 2*q; factors to:(m*20^q - 1) *(m*20^q + 1)odd n:factor of 5(Condition 2):All k where k = 6*m^2and m = = 1 or 4 mod 5:even n:factor of 5for odd n let k = 6*m^2and let n=2*q-1; factors to:[m*2^n*6^q - 1] *[m*2^n*6^q + 1] k = = 1 mod 1E (1E) 58 285, XE9, EX7, 1612, 18X6, 220X, 2553, 3124, 3420, 3427, 3658, 3E11, 3E53, 42X1, 4629, 5XE7, 6027, 620X, 6511, 67E8, 7172, 7239, 7819, 7E0E, 8837, 8927, 9052, 948X, 9748, 9X23, X110, X546, X81X, XE84, E090, E70X, EXX8, EE17, 100EE, 101X4, 10508, 10558, 10633, 10755, 11765, 125E2, 12668, 12805, 12839, 12863, 12X83, 13194, 13219, 13519, 1386X, 14255, 14514, 14841, 1484E, 1510E, 15141, 15151, 15E93, 16262, 16446, 16526, 16624, 166XE 5X77×20106433−16X82×20102245−111417×2010216X−110483×20102013−114636×20E685X−110058×20E3014−1 k = 2^2, 3^2, 7^2, 8^2. 10^2, 11^2, 15^2, 16^2 (etc. pattern repeating every 5m where k not = = 1 mod 1E) proven composite by condition 1.k = 6*1^2, 6*4^2, 6*6^2, 6*9^2, 6*E^2, 6*12^2, 6*14^2, 6*17^2 (etc. pattern repeating every Xm where k not = = 1 mod 1E) proven composite by condition 2.
1. this prime is GFN, if GFN are not counted, then the next prime would be 39×83+1
2. this prime is GFN
3. this prime is GFN, if GFN are not counted, then the next prime would be 2×181+1