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

361, 919, 1114, 1444

All k = m^2 for all n;

factors to:

(m*2^n - 1) *

(m*2^n + 1)

none - proven

659 (400258)

1211 (12621)

751 (6615)

674 (5838)

1159 (5628)

106 (4553)

1189 (3404)

1171 (2855)

373 (2508)

1103 (2203)

k = 1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2, 8^2, 9^2, 10^2, 11^2, 12^2, 13^2, 14^2, 15^2, 16^2, etc. (except 19^2) proven composite by full algebraic factors.

5

13, 17, 37, 41

none - proven

34 (163)

38 (28)

32 (8)

31 (5)

35 (4)

26 (4)

23 (4)

2 (4)

1 (3)

29 (2)

7

457, 1291, 3199, 3313

679, 691, 717, 859, 919, 1031, 1179, 1459, 1651, 1679, 1693, 1747, 1811, 1831, 1873, 1979, 2011, 2131, 2137, 2253, 2311, 2623, 2673, 2791, 2797, 2839, 2887, 3139, 3181, 3217, 3307 (k = 1 mod 6 at n=3K, other k at n=15K)

197 (181761)

367 (15118)

313 (5907)

1793 (5839)

159 (4896)

1469 (4669)

429 (3815)

3033 (2819)

2473 (2779)

2493 (2567)

8

14, 112, 116, 148

All k = m^3 for all n;

factors to:

(m*2^n - 1) *

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

none - proven

74 (2632)

37 (851)

142 (463)

73 (389)

92 (314)

127 (139)

104 (96)

47 (26)

84 (24)

43 (21)

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

9

41, 49, 74, 121

All k = m^2 for all n;

factors to:

(m*3^n - 1) *

(m*3^n + 1)

none - proven

119 (4486)

53 (536)

71 (23)

87 (15)

94 (12)

11 (11)

107 (9)

89 (8)

24 (8)

14 (8)

k = 1, 4, 9, 16, 25, 36, 64, 81, and 100 proven composite by full algebraic factors.

10

334, 1585, 1882, 3340

k = 343:

n = = 1 mod 3:

factor of 3

n = = 2 mod 3:

factor of 37

n = = 0 mod 3:

let n=3q; factors to:

(7*10^q - 1) * [49*10^(2q) + 7*10^q + 1]

2452 (554.7K)

1935 (51836)

1803 (45882)

1231 (37398)

1343 (29711)

505 (18470)

450 (11958)

3112 (3292)

1198 (2890)

2276 (2726)

2333 (2113)

11

5, 7, 17, 19

none - proven

1 (17)

9 (5)

16 (3)

15 (2)

14 (2)

8 (2)

3 (2)

2 (2)

18 (1)

13 (1)

12

376, 742, 1288, 1364

(Condition 1):

All k where k = m^2

and m = = 5 or 8 mod 13:

for even n let k = m^2

and let n = 2*q; factors to:

(m*12^q - 1) *

(m*12^q + 1)

odd n:

factor of 13

(Condition 2):

All k where k = 3*m^2

and m = = 3 or 10 mod 13:

even n:

factor of 13

for odd n let k = 3*m^2

and let n=2*q-1; factors to:

[m*2^(2q-1)*3^q - 1] *

[m*2^(2q-1)*3^q + 1]

none - proven

1132 (28717)

1037 (6281)

298 (1676)

1119 (1351)

1262 (1017)

534 (781)

844 (744)

943 (676)

647 (545)

831 (318)

k = 25, 64, 324, 441, 961, and 1156 proven composite by condition 1.

k = 27, 300, and 768 proven composite by condition 2.

13

29, 41, 69, 85

none - proven

43 (77)

76 (34)

52 (18)

25 (15)

28 (14)

20 (10)

34 (8)

72 (6)

47 (6)

42 (6)

14

4, 11, 19, 26

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*14^q - 1) *

(m*14^q + 1)

odd n:

factor of 5

none - proven

5 (19698)

2 (4)

1 (3)

24 (2)

23 (2)

20 (2)

17 (2)

15 (2)

8 (2)

25 (1)

k = 9 proven composite by partial algebraic factors.

16

100, 172, 211, 295

All k = m^2 for all n;

factors to:

(m*4^n - 1) *

(m*4^n + 1)

none - proven

74 (638)

137 (545)

178 (276)

148 (266)

273 (245)

219 (103)

152 (98)

235 (68)

203 (58)

263 (45)

k = 1, 4, 9, 16, 25, 36, 49, 64, 81, 121, 144, 169, 196, 225, 256, and 289 proven composite by full algebraic factors.

17

49, 59, 65, 86

All k where k = m^2

and m = = 7 or 9 mod 16:

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 2

none - proven

44 (6488)

29 (4904)

13 (1123)

36 (243)

10 (117)

26 (110)

5 (60)

11 (46)

58 (35)

46 (25)

k = 81 proven composite by partial algebraic factors.

18

246, 664, 723, 837

533, 597 (both at n=3K)

324 (25665)

628 (2213)

474 (1316)

457 (951)

501 (481)

337 (452)

151 (418)

811 (409)

711 (354)

261 (347)

19

9, 11, 29, 31

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*19^q - 1) *

(m*19^q + 1)

odd n:

factor of 5

none - proven

23 (108)

1 (19)

18 (6)

14 (6)

17 (5)

27 (4)

26 (3)

24 (2)

15 (2)

10 (2)

k = 4 proven composite by partial algebraic factors.

20

8, 13, 29, 34

none - proven

17 (22)

15 (21)

2 (10)

16 (9)

11 (8)

14 (6)

28 (3)

1 (3)

32 (2)

27 (2)

21

45, 65, 133, 153

none - proven

64 (2867)

131 (222)

101 (144)

47 (98)

29 (98)

84 (88)

142 (48)

109 (48)

61 (36)

77 (20)

23

5, 7, 17, 19

none - proven

14 (52)

3 (6)

2 (6)

4 (5)

1 (5)

18 (2)

15 (2)

12 (2)

11 (2)

8 (2)

25

105, 129, 211, 313

All k = m^2 for all n;

factors to:

(m*5^n - 1) *

(m*5^n + 1)

181, 235 (both at n=3K)

86 (1029)

268 (237)

177 (87)

274 (83)

170 (81)

137 (76)

265 (54)

58 (26)

272 (24)

130 (24)

k = 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, and 289 proven composite by full algebraic factors.

26

149, 334, 1892, 1987

178, 191, 223, 254, 284, 317, 355, 368, 380, 454, 562, 628, 729, 874, 892, 898, 926, 1061, 1147, 1153, 1157, 1184, 1189, 1204, 1208, 1214, 1279, 1312, 1375, 1376, 1541, 1549, 1657, 1736, 1774, 1852, 1901, 1930, 1934, 1945, 1963 (all at n=3K)

115 (520277)

32 (9812)

1094 (2586)

1186 (2541)

1237 (2277)

913 (1913)

1514 (1638)

121 (1509)

1704 (1486)

410 (1308)

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

23 (3742)

9 (23)

29 (13)

11 (10)

39 (8)

34 (8)

20 (8)

19 (8)

33 (7)

42 (5)

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

29

4, 9, 11, 13

none - proven

2 (136)

8 (38)

1 (5)

10 (3)

5 (2)

12 (1)

7 (1)

6 (1)

3 (1)

31

145, 265, 443, 493

5, 19, 51, 73, 97, 179, 191, 223, 235, 239, 247, 259, 274, 415, 421, 463, 467, 487, 489 (all at n=3K)

401 (2977)

123 (1872)

313 (1605)

209 (1589)

214 (1143)

124 (1116)

113 (643)

49 (637)

115 (464)

391 (378)

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)

29 (2M)

37 (6425)

13 (159)

44 (72)

26 (58)

54 (24)

39 (21)

47 (14)

3 (11)

53 (10)

42 (10)

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

33

545, 577, 764, 1633

(Condition 1):

All k where k = m^2

and m = = 4 or 13 mod 17:

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 17

(Condition 2):

All k where k = 33*m^2

and m = = 4 or 13 mod 17:

[Reverse condition 1]

(Condition 3):

All k where k = m^2

and m = = 15 or 17 mod 32:

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 2

257, 339, 817, 851, 951, 1123, 1240 (k = 257 and 339 at n=12K, other k at n=3K)

732 (19011)

186 (16770)

254 (3112)

562 (3087)

1408 (2920)

1157 (2647)

142 (2568)

1327 (1691)

1582 (1651)

370 (1628)

k = 16, 169, 441, 900, and 1444 proven composite by condition 1.

k = 528 proven composite by condition 2.

k = 225 and 289 proven composite by condition 3.

34

6, 29, 41, 64

none - proven

27 (3086)

31 (75)

44 (36)

59 (34)

57 (23)

33 (15)

1 (13)

20 (10)

22 (5)

21 (5)

35

5, 7, 17, 19

none - proven

1 (313)

3 (6)

2 (6)

16 (5)

14 (4)

8 (4)

15 (2)

11 (2)

6 (2)

18 (1)

37

29, 77, 113, 163

33, 149 (k = 33 at n=10K, k = 149 at n=3K)

81 (7683)

162 (1450)

56 (1158)

5 (900)

101 (574)

92 (313)

109 (188)

130 (146)

100 (99)

141 (74)

38

13, 14, 25, 53

44 (3K)

37 (136211)

22 (1579)

11 (766)

9 (43)

19 (41)

41 (26)

27 (23)

50 (16)

31 (15)

33 (9)

39

9, 11, 29, 31

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*39^q - 1) *

(m*39^q + 1)

odd n:

factor of 5

none - proven

1 (349)

14 (100)

24 (94)

16 (35)

30 (8)

15 (4)

13 (3)

27 (2)

23 (2)

19 (2)

k = 4 proven composite by partial algebraic factors.

41

8, 13, 17, 25

none - proven

14 (212)

7 (153)

5 (10)

23 (6)

18 (4)

11 (4)

22 (3)

1 (3)

20 (2)

6 (2)

43

21, 23, 65, 67

13, 55 (k = 13 at n=50K, k = 55 at n=3K)

53 (301)

4 (279)

35 (204)

12 (203)

61 (87)

17 (79)

59 (76)

39 (40)

31 (38)

3 (24)

44

4, 11, 19, 26

none - proven

23 (18)

8 (16)

24 (14)

20 (12)

16 (9)

12 (5)

1 (5)

2 (4)

25 (3)

21 (3)

45

93, 137, 277, 321

197, 257 (both at n=3K)

24 (153355)

53 (582)

286 (211)

205 (174)

70 (167)

313 (165)

106 (161)

102 (160)

29 (146)

204 (141)

47

5, 7, 13, 14

none - proven

4 (1555)

1 (127)

10 (51)

8 (32)

2 (4)

11 (2)

3 (2)

12 (1)

9 (1)

6 (1)

49

81, 129, 229, 241

All k = m^2 for all n;

factors to:

(m*7^n - 1) *

(m*7^n + 1)

82 (10K)

230 (24824)

194 (2530)

159 (2448)

139 (234)

79 (212)

115 (128)

216 (125)

44 (122)

106 (69)

209 (64)

k = 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225 proven composite by full algebraic factors.

50

16, 35, 67, 86

(Condition 1):

All k where k = m^2

and m = = 4 or 13 mod 17:

for even n let k = m^2

and let n = 2*q; factors to:

(m*50^q - 1) *

(m*50^q + 1)

odd n:

factor of 17

(Condition 2):

All k where k = 2*m^2

and m = = 3 or 14 mod 17:

even n:

factor of 17

for odd n let k = 2*m^2

and let n=2*q-1; factors to:

[m*5^(2q-1)*2^q - 1] *

[m*5^(2q-1)*2^q + 1]

37, 68 (k = 37 at n=121K, k = 68 at n=3K)

76 (1049)

14 (66)

49 (25)

73 (19)

52 (19)

13 (19)

84 (12)

5 (12)

75 (11)

44 (8)

No k's proven composite by condition 1.

k = 18 proven composite by condition 2.

51

25, 27, 77, 79

All k where k = m^2

and m = = 5 or 8 mod 13:

for even n let k = m^2

and let n = 2*q; factors to:

(m*51^q - 1) *

(m*51^q + 1)

odd n:

factor of 13

none - proven

1 (4229)

23 (96)

47 (40)

74 (23)

45 (21)

75 (10)

53 (9)

62 (8)

39 (8)

3 (8)

k = 64 proven composite by partial algebraic factors.

53

13, 17, 37, 41

none - proven

33 (1877)

22 (211)

12 (71)

10 (71)

2 (44)

29 (40)

26 (30)

38 (24)

14 (20)

16 (15)

54

21, 34, 76, 89

(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*54^q - 1) *

(m*54^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^q*3^(3q-1) - 1] *

[m*2^q*3^(3q-1) + 1]

45 (3K)

32 (1044)

87 (310)

23 (267)

59 (200)

82 (48)

46 (43)

26 (37)

79 (34)

44 (22)

72 (19)

k = 4, 9, 49, and 64 proven composite by condition 1.

k = 6 proven composite by condition 2.

55

13, 15, 41, 43

none - proven

3 (76)

22 (21)

1 (17)

27 (8)

11 (8)

23 (6)

20 (6)

39 (4)

19 (4)

9 (3)

56

20, 37, 77, 94

43 (3K)

59 (276)

83 (238)

76 (211)

29 (108)

65 (66)

74 (64)

36 (45)

38 (38)

26 (32)

88 (29)

57

144, 177, 233, 289

All k where k = m^2

and m = = 3 or 5 mod 8:

for even n let k = m^2

and let n = 2*q; factors to:

(m*57^q - 1) *

(m*57^q + 1)

odd n:

factor of 2

none - proven (with probable primes that have not been certified: k = 281)

281 (5610)

242 (1188)

87 (242)

262 (241)

278 (184)

204 (163)

54 (157)

201 (138)

173 (112)

100 (109)

k = 9, 25, 121, and 169 proven composite by partial algebraic factors.

59

4, 5, 7, 9

none - proven

3 (8)

1 (3)

8 (2)

2 (2)

6 (1)

61

125, 185, 373, 433

37, 53, 100, 139, 165, 229, 313, 353, 365, 389, 421 (k < 125 at n=10K, k > 125 at n=3K)

198 (41855)

404 (18637)

13 (4134)

77 (3080)

131 (2464)

430 (2248)

10 (1552)

406 (1289)

156 (1049)

41 (755)

62

8, 13, 29, 34

22, 26 (both at n=3K)

14 (80)

3 (59)

28 (51)

4 (9)

17 (6)

19 (5)

20 (4)

31 (3)

1 (3)

33 (2)

64

14, 51, 79, 116

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

(m*8^n - 1) *

(m*8^n + 1)

-or-

All k = m^3 for all n; factors to:

(m*4^n - 1) *

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

none - proven

24 (3020)

74 (1316)

106 (263)

92 (157)

99 (122)

69 (90)

104 (48)

85 (26)

47 (13)

84 (12)

k = 1, 4, 8, 9, 16, 25, 27, 36, 49, 64, 81, and 100 proven composite by full algebraic factors.

81

74, 575, 657, 737

All k = m^2 for all n;

factors to:

(m*9^n - 1) *

(m*9^n + 1)

123, 302, 477, 478, 630, 698, 716, 731 (all at n=3K)

581 (2403)

119 (2243)

487 (1405)

366 (1388)

443 (995)

513 (873)

241 (600)

327 (492)

546 (429)

662 (403)

k = 1, 4, 9, 16, 25, 36, 49, 64, 81, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, and 729 proven composite by full algebraic factors.

100

211, 235, 334, 750

(Condition 1):

All k = m^2 for all n;

factors to:

(m*10^n - 1) *

(m*10^n + 1)

(Condition 2):

k = 343:

n = = 1 mod 3:

factor of 37

n = = 2 mod 3:

factor of 3

n = = 0 mod 3:

let n=3q; factors to:

(7*100^q - 1) * [49*100^(2q) + 7*10^q + 1]

none - proven (with probable primes that have not been certified: k = 133, 469, and 505)

653 (717513)

74 (44709)

505 (9235)

450 (5979)

133 (5496)

469 (4451)

302 (2132)

470 (1957)

630 (1691)

690 (1310)

k = 1, 4, 9, 16, 25, 36, 49, 64, 81, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, and 729 proven composite by condition 1.

k = 343 proven composite by condition 2.

128

44, 59, 85, 86

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)

46 (142.857K)

29 (211192)

62 (44484)

23 (2118)

26 (1442)

74 (1128)

76 (759)

37 (699)

16 (459)

42 (246)

72 (124)

256

100, 172, 211, 295

All k = m^2 for all n;

factors to:

(m*16^n - 1) *

(m*16^n + 1)

191, 261, 286 (all at n=3K)

242 (37762)

262 (1856)

282 (948)

219 (471)

247 (336)

74 (319)

47 (228)

42 (224)

274 (148)

92 (143)

k = 1, 4, 9, 16, 25, 36, 49, 64, 81, 121, 144, 169, 196, 225, 256, and 289 proven composite by full algebraic factors.

512

14, 20, 37, 38

All k = m^3 for all n;

factors to:

(m*8^n - 1) *

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

none - proven

26 (3290)

4 (2215)

13 (2119)

24 (1655)

32 (472)

31 (93)

34 (51)

33 (40)

35 (34)

19 (21)

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

1024

81, 121, 124, 169

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

(m*32^n - 1) *

(m*32^n + 1)

-or-

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)

29, 31, 56, 61, 84, 91, 106, 109, 116, 136, 157, 166 (k = 29 at n=1M, other k at n=3K)

74 (666084)

39 (4070)

43 (2290)

99 (1226)

13 (1167)

151 (639)

78 (424)

105 (281)

114 (276)

137 (218)

k = 1, 4, 9, 16, 25, 32, 36, 49, 64, 100, and 144 proven composite by full algebraic factors.