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These are the k’s that make a full covering set with all or partial algebraic factors, thus excluded form the conjectures.

Sierpinski[]

base algebra factors the k’s (< CK)
8 All k = m^3 for all N; factors to:

(m*2^N + 1) * (m^2*4^N - m*2^N + 1)

1, 8
14 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)

1544, 1E194
23 All k = m^3 for all N; factors to:

(m*3^N + 1) * (m^2*9^N - m*3^N + 1)

8, 160, 368
28 All k = m^5 for all N; factors to:

(m*2^N + 1) * (m^4*14^N - m^3*8^N + m^2*4^N - m*2^N + 1)

1
47 k = 1544:

odd N:

factor of 7

N = = 2 mod 4:

factor of 15

N = = 0 mod 4:

let N=4q and let m=5*47^q; factors to:

(2*m^2 + 2m + 1) * (2*m^2 - 2m + 1)

1544
53 k = 1214368 & 9061000:

N = = 1 mod 3:

factor of 31

N = = 2 mod 3:

factor of 91

N = = 0 mod 3:

let N=3q and k=m^3; factors to:

(m*53^q + 1) * [m^2*53^(2q) - m*53^q + 1]

1214368, 9061000
54 All k = m^3 for all N; factors to:

(m*4^N + 1) * (m^2*14^N - m*4^N + 1)

1
69 All k = 4*q^4 for all N:

let k=4*q^4 and let m=q*3^N; factors to:

(2*m^2 + 2m + 1) * (2*m^2 - 2m + 1)

1544
X8 All k = m^7 for all N; factors to:

(m*2^N + 1) * (m^6*54^N - m^5*28^N + m^4*14^N - m^3*8^N + m^2*4^N - m*2^N + 1)

1

Riesel[]

base algebra factors the k’s (< CK)
4 All k = m^2 for all N; factors to:

(m*2^N - 1) * (m*2^N + 1)

3^2, 6^2, 9^2, 10^2, (etc. repeating every 3m)
9 All k = m^2 for all N; factors to:

(m*3^N - 1) * (m*3^N + 1)

4, 14, 30, 54
10 (Condition 1):

All k where k = m^2 and m = = 5 or 8 mod 11:

for even N let k = m^2 and 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^2 and m = = 3 or X mod 11:

even N:

factor of 11

for odd N let k = 3*m^2 and let N=2*q-1; factors to:

[m*2^N*3^q - 1] * [m*2^N*3^q + 1]

(Condition 1): 21, 54, 230

(Condition 2): 23, 210

14 All k = m^2 for all N; factors to:

(m*4^N - 1) * (m*4^N + 1)

3^2, 10^2, 13^2, 16^2, 23^2, 26^2, 29^2, 36^2, 39^2, 40^2, 49^2, 50^2, (etc. pattern repeating every 26m)
17 (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]

(Condition 1): 10^2, 16^2, 24^2, 36^2, 40^2, 4X^2 (etc. pattern repeating every 26m)

(Condition 2): 17*10^2, 17*16^2, 17*24*2, 17*36^2, 17*40^2, 17*4X^2 (etc. pattern repeating every 26m)

20 (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*20^q - 1) * (m*20^q + 1)

odd N:

factor of 5

(Condition 2):

All k where k = 6*m^2 and m = = 1 or 4 mod 5:

even N:

factor of 5

for odd N let k = 6*m^2 and let N=2*q-1; factors to:

[m*2^N*6^q - 1] * [m*2^N*6^q + 1]

(Condition 1): 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)

(Condition 2): 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 5m where k not = = 1 mod 1E)

21 All k = m^2 for all N; factors to:

(m*5^N - 1) * (m*5^N + 1)

6^2, 10^2, 16^2, 20^2, (etc. repeating every 6m)
23 All k = m^3 for all N; factors to:

(m*3^N - 1) * (m^2*9^N + m*3^N + 1)

8, 54, 160, 368
24 (Condition 1):

All k where k = m^2 and m = = 10 or 15 mod 25:

for even N let k = m^2 and let N = 2*q; factors to:

(m*24^q - 1) * (m*24^q + 1)

odd N:

factor of 25

(Condition 2):

All k where k = 7*m^2 and m = = 5 or 20 mod 25:

even N:

factor of 25

for odd N let k = 7*m^2 and let N=2*q-1; factors to:

[m*2^N*7^q - 1] * [m*2^N*7^q + 1]

(Condition 1): 100, 3309

(Condition 2): 2400

26 k = 961:

for even N let N=2*q; factors to:

(31*26^q - 1) * (31*26^q + 1)

odd N:

covering set 7, 11, 17

961
29 (Condition 1):

All k where k = m^2 and m = = 4 or 11 mod 15:

for even N let k = m^2 and let N = 2*q; factors to:

(m*29^q - 1) * (m*29^q + 1)

odd N:

factor of 15

(Condition 2):

All k where k = 29*m^2 and m = = 4 or 11 mod 15:

[Reverse condition 1]

(Condition 1): 14

(Condition 2): 380

30 All k = m^2 for all N; factors to:

(m*6^N - 1) * (m*6^N + 1)

2^2, 3^2, 5^2, 7^2, X^2, 10^2, 15^2, 16^2, 1E^2, 21^2, 24^2, 26^2, 28^2, 29^2, 2E^2, 31^2, 32^2, 34^2, 36^2, 39^2, 3E^2, 44^2, 45^2, 4X^2, 50^2, 53^2, 55^2, 57^2, 58^2, 5X^2, (etc. pattern repeating every 2Em)
33 (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*33^q - 1) * (m*33^q + 1)

odd N:

factor of 5

(Condition 2):

All k where k = 33*m^2 and m = = 2 or 3 mod 5:

[Reverse condition 1]

(Condition 1): 2^2, 8^2, 10^2, 1X^2, 24^2, 28^2, 32^2, 36^2, 40^2, (etc. pattern repeating every Xm where k not = = 1 mod 17)

(Condition 2): 33*2^2, 33*8^2, 33*10^2, 33*1X^2, 33*24^2, 33*28^2, 33*36^2, 33*40^2, (etc. pattern repeating every Xm where k not = = 1 mod 17)

34 (Condition 1):

All k where k = 34^s*m^2 and m = = 9 or 28 mod 35 and s is even and s>=0:

for even N let k = m^2 and let N = 2*q; factors to:

(m*34^q - 1) * (m*34^q + 1)

odd N:

factor of 35

(Condition 2):

All k where k = X*m^2 and m = = 16 or 1E mod 35:

even N:

factor of 35

for odd N let k = X*m^2 and let N=2*q-1; factors to:

[m*2^N*X^q - 1] * [m*2^N*X^q + 1]

(Condition 1): 9^2, 96^2, E0^2, 179^2, 193^2, 260^2, 343^2 (etc. pattern repeating every X3m where k not = = 1 mod 11)

(Condition 2): X*16^2, X*89^2, X*170^2, X*1X0^2, X*253^2, X*283^2, X*336^2, X*366^2

41 All k = m^2 for all N; factors to:

(m*7^N - 1) * (m*7^N + 1)

30, 100, 230, 400, 630, 900, 1030, 1400

General results[]

Both Sierpinski and Riesel bases[]

For r-th-power bases b (r is an odd number >1), all r-th-power ks proven composite by full algebra factors.

Only for Sierpinski bases[]

For fourth-power-bases b, all ks = 4m4 proven composite by full algebra factors.

Only for Riesel bases[]

For square bases b (r is an odd number >1), all square ks proven composite by full algebra factors.

These square ks proven composite by partial algebraic factors (algebra factors for even n, single factor for odd n), let k = m2: (also k = (b/q2)*m2 proven composite by partial algebraic factors (algebra factors for odd n, single factor for even n))

b m
4 mod 5 2 or 3 mod 5
10 mod 11 5 or 8 mod 11
14 mod 15 4 or 11 mod 15
24 mod 25 10 or 15 mod 25
30 mod 31 6 or 27 mod 31
34 mod 35 9 or 28 mod 35
44 mod 45 1E or 26 mod 45
50 mod 51 E or 42 mod 51
60 mod 61 23 or 3X mod 61
74 mod 75 2X or 47 mod 75
80 mod 81 1X or 63 mod 81
84 mod 85 X or 77 mod 85
90 mod 91 29 or 64 mod 91
94 mod 95 13 or 82 mod 95
E4 mod E5 31 or 84 mod E5
104 mod 105 38 or 89 mod 105
110 mod 111 24 or X9 mod 111
124 mod 125 68 or 79 mod 125
130 mod 131 17 or 116 mod 131
140 mod 141 69 or 94 mod 141
144 mod 145 12 or 133 mod 145
170 mod 171 8E or X2 mod 171
174 mod 175 75 or 100 mod 175
180 mod 181 54 or 129 mod 181
194 mod 195 14 or 181 mod 195
1X4 mod 1X5 6X or 137 mod 1X5
1E0 mod 1E1 50 or 161 mod 1E1
1E4 mod 1E5 45 or 170 mod 1E5
204 mod 205 E6 or 10E mod 205
220 mod 221 21 or 200 mod 221
224 mod 225 96 or 14E mod 225
240 mod 241 104 or 139 mod 241
250 mod 251 E4 or 159 mod 251
254 mod 255 36 or 21E mod 255
270 mod 271 88 or 1X5 mod 271
284 mod 285 97 or 1XX mod 285
290 mod 291 53 or 23X mod 291
294 mod 295 18 or 279 mod 295
2X0 mod 2X1 EE or 1X2 mod 2X1
2E0 mod 2E1 25 or 288 mod 2E1
300 mod 301 12E or 192 mod 301
314 mod 315 57 or 27X mod 315
320 mod 321 91 or 250 mod 321
324 mod 325 40 or 2X5 mod 325
364 mod 365 154 or 211 mod 365
374 mod 375 177 or 1EX mod 375
390 mod 391 44 or 349 mod 391
3X4 mod 3X5 9X or 307 mod 3X5
3E4 mod 3E5 72 or 343 mod 3E5
400 mod 401 20 or 3X1 mod 401
414 mod 415 65 or 370 mod 415
420 mod 421 X5 or 338 mod 421
430 mod 431 2E or 402 mod 431
434 mod 435 142 or 2E3 mod 435
454 mod 455 10X or 347 mod 455
464 mod 465 105 or 360 mod 465
470 mod 471 8X or 3X3 mod 471
480 mod 481 4X or 433 mod 481
484 mod 485 22 or 463 mod 485
4X4 mod 4X5 E3 or 3E2 mod 4X5
4E0 mod 4E1 80 or 431 mod 4E1
510 mod 511 255 or 278 mod 511
530 mod 531 73 or 47X mod 531
534 mod 535 33 or 502 mod 535
540 mod 541 52 or 4XE mod 541
544 mod 545 225 or 320 mod 545
564 mod 565 15E or 406 mod 565
574 mod 575 226 or 34E mod 575
584 mod 585 207 or 37X mod 585
590 mod 591 186 or 407 mod 591
5E0 mod 5E1 239 or 374 mod 5E1
5E4 mod 5E5 153 or 462 mod 5E5
610 mod 611 107 or 506 mod 611
614 mod 615 283 or 352 mod 615
654 mod 655 230 or 425 mod 655
660 mod 661 144 or 519 mod 661
664 mod 665 81 or 5X4 mod 665
674 mod 675 30X or 367 mod 675
694 mod 695 190 or 505 mod 695
6E0 mod 6E1 115 or 598 mod 6E1
700 mod 701 331 or 390 mod 701
704 mod 705 39 or 688 mod 705
710 mod 711 272 or 45E mod 711
720 mod 721 257 or 486 mod 721
734 mod 735 2E6 or 43E mod 735
744 mod 745 87 or 67X mod 745
750 mod 751 189 or 584 mod 751
770 mod 771 382 or 3XE mod 771
774 mod 775 245 or 530 mod 775
784 mod 785 256 or 52E mod 785
790 mod 791 15X or 633 mod 791
7X0 mod 7X1 120 or 681 mod 7X1
800 mod 801 E8 or 705 mod 801
824 mod 825 183 or 662 mod 825
834 mod 835 136 or 6EE mod 835
840 mod 841 41 or 800 mod 841
850 mod 851 353 or 4EX mod 851
854 mod 855 66 or 7XE mod 855
864 mod 865 419 or 448 mod 865
870 mod 871 396 or 497 mod 871
880 mod 881 409 or 474 mod 881
8X4 mod 8X5 95 or 810 mod 8X5
8E4 mod 8E5 33E or 576 mod 8E5
900 mod 901 30 or 891 mod 901
904 mod 905 43 or 882 mod 905
920 mod 921 195 or 748 mod 921
954 mod 955 432 or 523 mod 955
964 mod 965 478 or 4X9 mod 965
970 mod 971 266 or 707 mod 971
994 mod 995 318 or 679 mod 995
9E0 mod 9E1 438 or 575 mod 9E1
9E4 mod 9E5 392 or 623 mod 9E5
X10 mod X11 355 or 678 mod X11
X34 mod X35 329 or 708 mod X35
X40 mod X41 169 or 894 mod X41
X44 mod X45 300 or 745 mod X45
X90 mod X91 74 or X19 mod X91
X94 mod X95 243 or 852 mod X95
E10 mod E11 42X or 6X3 mod E11
E14 mod E15 34 or XX1 mod E15
E20 mod E21 377 or 766 mod E21
E24 mod E25 X7 or X3X mod E25
E30 mod E31 11X or X13 mod E31
E44 mod E45 224 or 921 mod E45
E60 mod E61 553 or 60X mod E61
E70 mod E71 164 or X09 mod E71
E90 mod E91 78 or E15 mod E91
E94 mod E95 2X6 or 8XE mod E95
EX4 mod EX5 286 or 91E mod EX5
EE4 mod EE5 335 or 880 mod EE5
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