Base

Conjectured first 4 Riesel k

k's that make a full covering set with all or partial algebraic factors

Remaining k to find prime

(n testing limit)

Top 10 k's with largest first primes: k (n)

(sorted by n only)

Comments

4

419, 659, 794, 1466

1238, 1286 (both at n=3K)

186 (10458)

766 (3196)

839 (1217)

1194 (1075)

1206 (590)

1221 (586)

1301 (574)

1201 (480)

857 (471)

1154 (449)

5

7, 11, 31, 35

none - proven

34 (8)

27 (4)

22 (4)

19 (4)

32 (3)

18 (3)

28 (2)

25 (2)

24 (2)

16 (2)

7

209, 1463, 3305, 3533

389, 443, 563, 569, 687, 689, 827, 1009, 1049, 1101, 1467, 1511, 1604, 1627, 1787, 1789, 2083, 2089, 2183, 2369, 2401, 2441, 2507, 2627, 2663, 2867, 2921, 3047, 3209, 3289, 3428, 3489 (all at n=3K)

2786 (2853)

2801 (2688)

1775 (2625)

1921 (2556)

429 (2362)

3503 (2337)

1087 (2331)

1367 (2267)

1927 (2147)

863 (1985)

8

47, 79, 83, 181

All k = m^3 for all n;

factors to:

(m*2^n - 1) *

(m^2*4^n + m*2^n + 1)

none - proven

173 (7771)

118 (820)

94 (194)

68 (115)

143 (111)

169 (94)

155 (87)

134 (71)

139 (58)

91 (56)

k = 1, 8, 27, 64, and 125 proven composite by full algebraic factors.

9

31, 39, 111, 119

none - proven

41 (2446)

94 (57)

76 (26)

74 (25)

61 (22)

83 (13)

52 (11)

103 (10)

56 (10)

114 (9)

10

989, 1121, 3653, 3662

100, 269, 1343, 2573 (k = 100 at n=2.147G, k = 269 at n=100K, other k at n=3K)

804 (5470)

3356 (4584)

1024 (4554)

2157 (3560)

2661 (2681)

1844 (1643)

1475 (1128)

2683 (1049)

2311 (1000)

3301 (788)

11

5, 7, 17, 19

none - proven

10 (10)

16 (8)

13 (2)

12 (2)

9 (2)

4 (2)

1 (2)

18 (1)

15 (1)

14 (1)

12

521, 597, 1143, 1509

12, 885, 911, 976, 1041, 1433, 1468 (k = 12 at n=33.55M, other k < 1143 at n=25K, k > 1143 at n=3K)

404 (714558)

1052 (5715)

563 (4020)

1431 (2784)

378 (2388)

846 (1384)

1197 (996)

1247 (751)

1057 (690)

261 (644)

13

15, 27, 47, 71

none - proven (with probable primes that have not been certified: k = 29)

29 (10574)

48 (6267)

67 (570)

11 (564)

51 (76)

59 (42)

50 (34)

19 (30)

64 (26)

68 (15)

14

4, 11, 19, 26

none - proven

22 (16)

10 (6)

6 (6)

25 (4)

21 (4)

23 (3)

18 (2)

16 (2)

13 (2)

7 (2)

16

38, 194, 524, 608

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)

89 (25K)

186 (5229)

459 (3701)

215 (3373)

530 (1210)

23 (1074)

515 (940)

230 (863)

398 (589)

154 (186)

493 (170)

k = 4, 64, and 324 proven composite by full algebraic factors.

17

31, 47, 127, 143

53 (3K)

92 (51311)

88 (4868)

10 (1356)

104 (871)

137 (723)

121 (684)

103 (254)

128 (225)

7 (190)

106 (144)

18

398, 512, 571, 989

18, 607, 761, 873, 922, 983 (k = 18 at n=16.77M, other k at n=3K)

122 (292318)

381 (24108)

291 (2415)

37 (457)

523 (440)

987 (409)

547 (317)

457 (262)

362 (258)

123 (236)

19

9, 11, 29, 31

none - proven

5 (78)

18 (29)

15 (17)

17 (14)

6 (14)

21 (6)

16 (6)

26 (4)

28 (3)

4 (3)

20

8, 13, 29, 34

none - proven

22 (106)

6 (15)

19 (14)

17 (13)

25 (10)

16 (8)

28 (4)

15 (4)

11 (3)

31 (2)

21

23, 43, 47, 111

none - proven

67 (2490)

107 (422)

89 (204)

78 (16)

88 (10)

12 (10)

98 (9)

27 (8)

72 (7)

55 (7)

23

5, 7, 17, 19

none - proven

8 (119215)

10 (3762)

4 (342)

13 (152)

18 (7)

14 (5)

16 (4)

1 (4)

3 (3)

9 (2)

25

79, 103, 185, 287

71, 181, 222 (k = 71 at n=10K, k = 181 at n=5K, k = 222 at n=350K)

61 (3104)

89 (815)

40 (518)

209 (376)

80 (304)

119 (198)

223 (187)

283 (186)

194 (117)

151 (112)

26

221, 284, 1627, 1766

65, 155, 293, 314, 430, 470, 536, 569, 620, 676, 746, 825, 850, 1018, 1025, 1051, 1124, 1152, 1153, 1172, 1229, 1240, 1262, 1270, 1300, 1481, 1531, 1669, 1693, 1750, 1756, 1762 (k = 65 and 155 at n=1M, other k at n=3K)

607 (1089034)

32 (318071)

640 (66510)

715 (38694)

307 (37834)

217 (11454)

373 (6886)

283 (6882)

998 (2423)

398 (2393)

27

13, 15, 41, 43

All k = m^3 for all n;

factors to:

(m*3^n - 1) *

(m^2*9^n + m*3^n + 1)

none - proven (with probable primes that have not been certified: k = 33)

33 (7876)

21 (112)

29 (54)

39 (13)

9 (10)

42 (7)

40 (5)

36 (4)

26 (4)

22 (4)

k = 1, 8, and 27 proven composite by full algebraic factors.

29

4, 7, 11, 19

none - proven

13 (6)

10 (4)

6 (4)

17 (3)

14 (3)

16 (2)

3 (2)

1 (2)

18 (1)

15 (1)

31

239, 293, 521, 1025

1, 51, 73, 77, 107, 117, 149, 181, 209, 241, 263, 299, 349, 365, 389, 404, 417, 435, 461, 469, 491, 493, 537, 543, 545, 569, 579, 596, 619, 635, 649, 657, 677, 689, 723, 753, 789, 811, 829, 857, 879, 967 (k = 1 at n=524K, other k < 239 at n=6K, other k at n=3K)

43 (21053)

189 (5570)

674 (2849)

743 (2575)

665 (2366)

581 (2134)

1019 (2091)

551 (2087)

431 (1981)

933 (1830)

32

10, 23, 43, 56

All k = m^5 for all n; factors to:

(m*2^n + 1) *

(m^4*16^n - m^3*8^n + m^2*4^n - m*2^n + 1)

4, 16 (k = 4 at n=1.717G, k = 16 at n=3.435G)

47 (1223)

26 (63)

55 (44)

41 (43)

48 (37)

52 (16)

9 (13)

45 (12)

31 (12)

34 (10)

k = 1 and 32 proven composite by full algebraic factors.

33

511, 543, 1599, 1631

67, 203, 1207, 1317, 1439, 1531, 1563, 1597 (k = 67 and 203 at n=12K, other k at n=3K)

766 (610412)

36 (23615)

1188 (23614)

407 (10961)

1580 (9213)

1240 (6953)

154 (6846)

596 (6244)

319 (5043)

288 (4583)

34

6, 29, 41, 64

none - proven

11 (310)

51 (42)

24 (31)

26 (28)

59 (23)

61 (16)

57 (12)

5 (12)

16 (8)

62 (7)

35

5, 7, 17, 19

none - proven

4 (42)

9 (4)

16 (2)

13 (2)

10 (2)

1 (2)

18 (1)

15 (1)

14 (1)

12 (1)

37

39, 75, 87, 191

37, 63, 94, 127, 134, 171 (k = 37 at n=524K, k = 94 at n=1M, other k at n=3K)

19 (5310)

52 (1628)

139 (1526)

18 (461)

113 (389)

181 (245)

72 (120)

133 (77)

143 (69)

179 (66)

38

14, 16, 25, 53

1 (16.77M)

2 (2729)

31 (1528)

51 (143)

34 (62)

9 (21)

52 (16)

37 (16)

24 (10)

4 (10)

46 (8)

39

9, 11, 29, 31

none - proven

19 (831)

17 (10)

26 (8)

27 (5)

20 (5)

10 (4)

25 (2)

21 (2)

16 (2)

13 (2)

41

8, 13, 15, 23

none - proven

22 (68)

1 (16)

4 (6)

16 (4)

14 (3)

6 (3)

19 (2)

12 (2)

10 (2)

7 (2)

43

21, 23, 65, 67

37, 56 (both at n=3K)

31 (833)

13 (580)

9 (498)

3 (171)

5 (38)

63 (35)

17 (34)

54 (27)

41 (24)

15 (23)

44

4, 11, 19, 26

none - proven

25 (382)

24 (59)

17 (47)

1 (16)

5 (15)

3 (9)

21 (8)

10 (6)

22 (4)

12 (3)

45

47, 91, 231, 275

139, 217 (both at n=3K)

24 (18522)

235 (1832)

114 (1439)

93 (1180)

149 (615)

243 (276)

75 (230)

59 (184)

187 (150)

212 (95)

47

5, 7, 8, 16

none - proven

2 (175)

13 (40)

1 (8)

11 (5)

12 (3)

10 (2)

9 (2)

4 (2)

15 (1)

14 (1)

49

31, 79, 179, 191

none - proven

186 (33764)

183 (1610)

96 (662)

141 (522)

43 (176)

24 (165)

121 (126)

101 (108)

167 (102)

94 (91)

50

16, 35, 67, 86

1 (16.77M)

37 (956)

33 (683)

7 (516)

59 (135)

42 (80)

79 (72)

44 (53)

31 (28)

69 (22)

58 (14)

51

25, 27, 77, 79

none - proven

38 (4881)

61 (310)

52 (183)

45 (182)

41 (80)

66 (74)

48 (32)

73 (18)

54 (12)

40 (12)

53

7, 11, 31, 35

4, 17, 19 (k = 4 at n=2M, other k at n=3K)

8 (227183)

6 (143)

13 (28)

30 (18)

10 (16)

27 (14)

18 (11)

5 (9)

1 (8)

25 (4)

54

21, 34, 76, 89

none - proven

74 (311)

19 (103)

81 (102)

36 (86)

16 (30)

66 (20)

61 (16)

60 (16)

53 (15)

67 (12)

55

13, 15, 41, 43

1, 36 (k = 1 at n=524K, k = 36 at n=1M)

22 (92)

33 (54)

21 (18)

30 (12)

10 (9)

37 (4)

39 (3)

40 (2)

38 (2)

29 (2)

56

20, 37, 77, 94

46 (3K)

39 (394)

35 (243)

85 (100)

4 (78)

19 (70)

34 (34)

26 (27)

68 (21)

30 (19)

76 (18)

57

47, 175, 231, 311

117, 207 (both at n=3K)

150 (15759)

14 (14955)

181 (1319)

201 (189)

287 (187)

303 (133)

236 (125)

291 (104)

163 (98)

217 (95)

59

4, 5, 7, 9

none - proven

8 (5)

2 (3)

6 (2)

1 (2)

3 (1)

61

63, 123, 311, 371

119, 127, 155, 164, 230, 249, 262, 324, 340, 342, 353, 359, 368 (all at n=3K)

62 (3698)

43 (2788)

152 (2479)

114 (1706)

23 (1659)

309 (773)

295 (719)

165 (286)

135 (267)

222 (238)

62

8, 13, 29, 34

1, 27 (k = 1 at n=16.77M, k = 27 at n=3K)

16 (2580)

7 (308)

22 (248)

32 (139)

14 (67)

10 (66)

2 (43)

25 (40)

12 (27)

33 (22)

64

14, 51, 79, 116

All k = m^3 for all n;

factors to:

(m*4^n - 1) *

(m^2*16^n + m*4^n + 1)

none - proven

11 (3222)

89 (1269)

94 (97)

98 (42)

86 (36)

24 (31)

91 (28)

101 (22)

84 (21)

92 (14)

k = 1, 8, 27, and 64 proven composite by full algebraic factors.

81

575, 649, 655, 1167

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)

239, 335, 514, 698, 964, 1007 (k < 575 at n=5K, other k at n=3K)

558 (51992)

778 (14903)

311 (7834)

621 (5205)

75 (3309)

569 (2937)

439 (2097)

284 (1455)

821 (1378)

831 (1337)

k = 4, 64, 324, and 1024 proven composite by full algebraic factors.

100

62, 233, 332, 836

100, 269, 335, 433 (k = 100 at n=1.073G, k = 269 at n=50K, k = 335 at n=3K, k = 433 at n=1M)

684 (563559)

64 (529397)

75 (16392)

591 (13007)

594 (2932)

804 (2735)

563 (1044)

764 (807)

349 (784)

712 (711)

128

44, 85, 98, 173

All k = m^7 for all n;

factors to:

(m*2^n + 1) *

(m^6*64^n - m^5*32^n + m^4*16^n - m^3*8^n + m^2*4^n - m*2^n + 1)

16, 40, 47, 83, 88, 94, 122 (k = 16 at n=4.908G, other k at n=1.2857M)

171 (102533)

118 (44110)

41 (39271)

158 (17403)

52 (15608)

42 (13001)

146 (4567)

156 (1083)

136 (1044)

172 (760)

k = 1 proven composite by full algebraic factors.

k = 8 and 32 have no possible prime.

256

38, 194, 467, 524

All k=4*q^4 for all n:

let k=4*q^4

and let m=q*4^n; factors to:

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

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

89, 116, 215, 230, 281, 329, 383, 398, 407, 434, 459, 504 (all at n=3K)

11 (5702)

346 (2914)

449 (2518)

263 (1914)

523 (1428)

368 (1354)

51 (1198)

309 (1088)

180 (768)

129 (652)

k = 4, 64, and 324 proven composite by full algebraic factors.

512

18, 20, 37, 47

All k = m^3 for all n;

factors to:

(m*8^n + 1) *

(m^2*64^n - m*8^n + 1)

2, 4, 5, 16, 32, 34 (k = 2 at n=2.001P, k = 4 at n=62.54T, k = 5 at n=1M, k = 16 at n=1.954T, k = 32 at n=3.817G, k = 34 at n=3K)

25 (1058)

19 (1050)

42 (44)

31 (24)

44 (23)

12 (23)

14 (21)

7 (20)

11 (9)

46 (8)

k = 1, 8, and 27 proven composite by full algebraic factors.

1024

81, 124, 286, 329

All k = m^5 for all n;

factors to:

(m*4^n + 1) *

(m^4*256^n - m^3*64^n + m^2*16^n - m*4^n + 1)

4, 16, 29, 38, 56, 94, 101, 119, 139, 152, 164, 176, 194, 199, 208, 219, 227, 239, 242, 245, 247, 254, 256, 269, 281, 326 (all at n=3K)

174 (189139)

321 (44248)

171 (29984)

114 (8969)

111 (6130)

96 (4266)

138 (4069)

44 (1933)

187 (1779)

304 (1703)

k = 1, 32, and 243 proven composite by full algebraic factors.