Base

Conjectured smallest Riesel k

Covering set

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

2

509203

3, 5, 7, 13, 17, 241

23669, 31859, 38473, 46663, 67117, 74699, 81041, 93839, 97139, 107347, 121889, 129007, 143047, 161669, 206231, 215443, 226153, 234343, 245561, 250027, 315929, 319511, 324011, 325123, 327671, 336839, 342847, 344759, 351134, 362609, 363343, 364903, 365159, 368411, 371893, 384539, 386801, 397027, 409753, 444637, 470173, 474491, 477583, 478214, 485557, 494743 (k = 351134 and 478214 at n=8M, other k at n=12.5M)

192971 (14773498)

206039 (13104952)

2293 (12918431)

9221 (11392194)

146561 (11280802)

273809 (8932416)

502573 (7181987)

402539 (7173024)

40597 (6808509)

304207 (6643565)

3

12119

2, 5, 7, 13, 73

1613, 1831, 1937, 3131, 3589, 5755, 6787, 7477, 7627, 7939, 8713, 8777, 9811, 10651, 11597 (all at n=50K)

8059 (47256)

11753 (36665)

6119 (28580)

7511 (26022)

313 (24761)

11251 (24314)

9179 (21404)

997 (20847)

6737 (17455)

7379 (16856)

4

361

3, 5, 7, 13

All k = m^2 for all n;

factors to:

(m*2^n - 1) *

(m*2^n + 1)

none - proven

106 (4553)

74 (1276)

219 (206)

191 (113)

312 (51)

247 (42)

223 (33)

274 (22)

234 (18)

91 (17)

k = 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, and 324 proven composite by full algebraic factors.

5

13

2, 3

none - proven

2 (4)

1 (3)

11 (2)

8 (2)

12 (1)

9 (1)

7 (1)

6 (1)

4 (1)

3 (1)

6

84687

7, 13, 31, 37, 97

1597, 6236, 9491, 37031, 49771, 50686, 53941, 55061, 57926, 76761, 79801, 83411 (k = 1597 at n=5.6M, other k at n=40K)

36772 (1723287)

43994 (569498)

77743 (560745)

51017 (528803)

57023 (483561)

78959 (458114)

59095 (171929)

48950 (143236),

29847 (141526)

9577 (121099)

7

457

2, 3, 5, 13, 19

none - proven (with probable primes that have not been certified: k = 197)

197 (181761)

367 (15118)

313 (5907)

159 (4896)

429 (3815)

419 (1052)

391 (938)

299 (600)

139 (468)

79 (424)

8

14

3, 5, 13

All k = m^3 for all n;

factors to:

(m*2^n - 1) *

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

none - proven

11 (18)

5 (4)

12 (3)

7 (3)

2 (2)

13 (1)

10 (1)

9 (1)

6 (1)

4 (1)

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

9

41

2, 5

All k = m^2 for all n;

factors to:

(m*3^n - 1) *

(m*3^n + 1)

none - proven

11 (11)

24 (8)

14 (8)

38 (3)

18 (3)

39 (2)

34 (2)

32 (2)

29 (2)

27 (2)

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

10

334

3, 7, 13, 37

none - proven

121 (483)

109 (136)

98 (90)

230 (60)

289 (35)

89 (33)

32 (28)

233 (18)

324 (17)

100 (17)

11

5

2, 3

none - proven

1 (17)

3 (2)

2 (2)

4 (1)

12

376

5, 13, 29

(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

298 (1676)

157 (285)

46 (194)

304 (40)

259 (40)

94 (36)

292 (30)

147 (28)

301 (27)

349 (25)

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

k = 27 and 300 proven composite by condition 2.

13

29

2, 7

none - proven

25 (15)

28 (14)

20 (10)

1 (5)

22 (3)

17 (3)

16 (3)

27 (2)

21 (2)

12 (2)

14

4

3, 5

none - proven

2 (4)

1 (3)

3 (1)

15

622403

2, 17, 113, 1489

47, 203, 239, 407, 437, 451, 889, 893, 1945, 2049, 2245, 2487, 2507, 2689, 2699, 2863, 3059, 3163, 3179, 3261, 3409, 3697, 3701, 3725, 4173, 4249, 4609, 4771, 4877, 5041, 5243, 5425, 5441, 5503, 5669, 5857, 5913, 5963, 6231, 6447, 6787, 6879, 6999, 7386, 7407, 7459, 7473, 7527, 7615, 7683, 7687, 7859, 8099, 8621, 8671, 8839, 8863, 9025, 9267, 9409, 9655, 9663, 9707, 9817, 9955 (for k <= 10K) (all at n=1.5K)

2940 (13254)

8610 (5178)

2069 (1461)

3917 (1427)

1145 (1349)

1583 (1330)

7027 (1316)

8831 (1296)

5305 (1273)

4865 (1265)

16

100

3, 7, 13

All k = m^2 for all n;

factors to:

(m*4^n - 1) *

(m*4^n + 1)

none - proven

74 (638)

78 (26)

48 (15)

58 (12)

31 (12)

95 (8)

46 (8)

88 (6)

44 (6)

39 (6)

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

17

49

2, 3

none - proven

44 (6488)

29 (4904)

13 (1123)

36 (243)

10 (117)

26 (110)

5 (60)

11 (46)

46 (25)

35 (24)

18

246

5, 13, 19

none - proven

151 (418)

78 (172)

50 (110)

79 (63)

237 (44)

184 (44)

75 (44)

215 (36)

203 (32)

93 (32)

19

9

2, 5

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

1 (19)

7 (2)

3 (2)

8 (1)

6 (1)

5 (1)

2 (1)

k = 4 proven composite by partial algebraic factors.

20

8

3, 7

none - proven

2 (10)

1 (3)

6 (2)

5 (2)

7 (1)

4 (1)

3 (1)

21

45

2, 11

none - proven

29 (98)

34 (17)

43 (10)

32 (4)

5 (4)

6 (3)

1 (3)

44 (2)

37 (2)

31 (2)

22

2738

5, 23, 97

208, 211, 898, 976, 1036, 1885, 1933, 2050, 2161, 2278, 2347, 2434 (all at n=13K)

1013 (26067)

185 (11433)

1335 (11155)

2719 (9671)

2083 (8046)

883 (5339)

2529 (3700)

2116 (3371)

2230 (3236)

1119 (2849)

23

5

2, 3

none - proven

3 (6)

2 (6)

4 (5)

1 (5)

24

32336

5, 7, 13, 73, 577

(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*24^q - 1) *

(m*24^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^(3q-1)*3^q - 1] *

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

389, 461, 1581, 1711, 2094, 2606, 3006, 3754, 4239, 5356, 5784, 5791, 6116, 6579, 6781, 6831, 7321, 7809, 10219, 10399, 10666, 11101, 11516, 12326, 12429, 12674, 13269, 13691, 15019, 15151, 15614, 15641, 16124, 16234, 16616, 17019, 17436, 18054, 18454, 18964, 19116, 20026, 20576, 20611, 20879, 21004, 21464, 21524, 21639, 21809, 23549, 24404, 25046, 25136, 25349, 25389, 25419, 25646, 25731, 26176, 26229, 26661, 27049, 27154, 28001, 28384, 28849, 28859, 29211, 29531, 29569, 29581, 31071, 31466, 31734, 31854, 31994, 31996, 32099 (k = 1 mod 23 at n=12.4K, other k at n=260K)

10171 (259815)

11906 (252629)

23059 (252514)

21411 (252303)

28554 (239686)

20804 (233296)

8894 (210624)

2844 (203856)

25379 (175842)

22604 (169372)

k = 2^2, 3^2, 7^2, 8^2, 12^2, 13^2, 17^2, 18^2 (etc. pattern repeating every 5m) proven composite by condition 1.

k = 6*1^2, 6*4^2, 6*6^2, 6*9^2, 6*11^2, 6*14^2, 6*16^2, 6*19^2 (etc. pattern repeating every 5m) proven composite by condition 2.

25

105

2, 13

All k = m^2 for all n;

factors to:

(m*5^n - 1) *

(m*5^n + 1)

none - proven

86 (1029)

58 (26)

72 (24)

67 (24)

79 (21)

37 (17)

38 (14)

92 (13)

57 (10)

98 (9)

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

26

149

3, 7, 31, 37

none - proven

115 (520277)

32 (9812)

121 (1509)

73 (537)

80 (382)

128 (300)

124 (249)

37 (233)

25 (133)

65 (100)

27

13

2, 7

All k = m^3 for all n;

factors to:

(m*3^n - 1) *

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

none - proven

9 (23)

11 (10)

12 (2)

7 (2)

6 (2)

3 (2)

10 (1)

5 (1)

4 (1)

2 (1)

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

28

3769

5, 29, 157

(Condition 1):

All k where k = m^2

and m = = 12 or 17 mod 29:

for even n let k = m^2

and let n = 2*q; factors to:

(m*28^q - 1) *

(m*28^q + 1)

odd n:

factor of 29

(Condition 2):

All k where k = 7*m^2

and m = = 5 or 24 mod 29:

even n:

factor of 29

for odd n let k = 7*m^2

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

[m*2^(2q-1)*7^q - 1] *

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

233, 376, 943, 1132, 1422, 2437 (k = 233 and 1422 at n=1M, other k at n=20.3K)

2319 (65184)

3232 (9147)

3019 (7073)

460 (5400)

1688 (4760)

2406 (4634)

2464 (4324)

849 (3129)

1507 (2938)

472 (2414)

k = 144, 289, 1681, and 2116 proven composite by condition 1.

k = 175 proven composite by condition 2.

29

4

3, 5

none - proven

2 (136)

1 (5)

3 (1)

30

4928

13, 19, 31, 67

k = 1369:

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

(37*30^q - 1) *

(37*30^q + 1)

odd n:

covering set 7, 13, 19

659, 1024, 1580, 1936, 2293, 2916, 3719, 4372, 4897 (all at n=500K)

1642 (346592)

239 (337990)

2538 (262614)

249 (199355)

3256 (160619)

225 (158755)

774 (148344)

1873 (50427)

3253 (43291)

1654 (38869)

31

145

2, 3, 7, 19

5, 19, 51, 73, 97 (all at n=6K)

123 (1872)

124 (1116)

113 (643)

49 (637)

115 (464)

21 (275)

39 (250)

70 (149)

142 (140)

33 (107)

32

10

3, 11

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)

none - proven

3 (11)

2 (6)

9 (3)

8 (2)

5 (2)

7 (1)

6 (1)

4 (1)

k = 1 proven composite by full algebraic factors.

33

545

2, 17

(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 (both at n=12K)

186 (16770)

254 (3112)

142 (2568)

370 (1628)

272 (1418)

222 (919)

108 (360)

213 (233)

387 (191)

277 (187)

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

k = 528 proven composite by condition 2.

k = 225 and 289 proven composite by condition 3.

34

6

5, 7

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*34^q - 1) *

(m*34^q + 1)

odd n:

factor of 5

none - proven

1 (13)

5 (2)

3 (1)

2 (1)

k = 4 proven composite by partial algebraic factors.

35

5

2, 3

none - proven

1 (313)

3 (6)

2 (6)

4 (1)

36

33791

13, 31, 43, 97

All k = m^2 for all n;

factors to:

(m*6^n - 1) *

(m*6^n + 1)

1148, 1555, 2110, 2133, 3699, 4551, 4737, 6236, 6883, 7253, 7362, 7399, 7991, 8250, 8361, 8363, 8472, 9491, 9582, 11014, 12320, 12653, 13641, 14358, 14540, 14836, 14973, 14974, 15228, 15687, 15756, 15909, 16168, 17354, 17502, 17946, 18203, 19035, 19646, 20092, 20186, 20630, 21880, 22164, 22312, 23213, 23901, 23906, 24236, 24382, 24645, 24731, 24887, 25011, 25159, 25161, 25204, 25679, 25788, 26160, 26355, 27161, 29453, 29847, 30970, 31005, 31634, 32302, 33047, 33627 (all at n=10K)

13800 (9790)

20485 (9140)

19389 (9119)

20684 (8627)

19907 (8439)

11216 (7524)

28416 (7315)

32380 (7190)

27296 (7115)

10695 (6672)

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. proven composite by full algebraic factors.

37

29

2, 5, 7, 13, 67

none - proven

5 (900)

19 (63)

18 (14)

1 (13)

8 (4)

25 (3)

23 (3)

14 (3)

6 (3)

4 (3)

38

13

3, 5, 17

none - proven

11 (766)

9 (43)

7 (7)

1 (3)

12 (2)

8 (2)

5 (2)

2 (2)

10 (1)

6 (1)

39

9

2, 5

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)

7 (2)

3 (2)

2 (2)

8 (1)

6 (1)

5 (1)

k = 4 proven composite by partial algebraic factors.

40

25462

3, 7, 41, 223

(Condition 1):

All k where k = m^2

and m = = 9 or 32 mod 41:

for even n let k = m^2

and let n = 2*q; factors to:

(m*40^q - 1) *

(m*40^q + 1)

odd n:

factor of 41

(Condition 2):

All k where k = 10*m^2

and m = = 18 or 23 mod 41:

even n:

factor of 41

for odd n let k = 10*m^2

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

[m*2^(3q-1)*5^q - 1] *

[m*2^(3q-1)*5^q + 1]

157, 534, 618, 709, 739, 787, 862, 1067, 1114, 1174, 1559, 1805, 2254, 2887, 3418, 3650, 4006, 4582, 4673, 4771, 6107, 6463, 6682, 6684, 6946, 7094, 7258, 7282, 7381, 7504, 7702, 7795, 8035, 8461, 8572, 9226, 9347, 9472, 9716, 9748, 9964, 10285, 10615, 10744, 11030, 11470, 11479, 11560, 11847, 12178, 12193, 12250, 12299, 12301, 12568, 12742, 13005, 13022, 13039, 13191, 13624, 13666, 13777, 13939, 14146, 14262, 14494, 15374, 15417, 15496, 15661, 15730, 16579, 16705, 16891, 16932, 17014, 17275, 17344, 17923, 17998, 18949, 19117, 19310, 19606, 19722, 19761, 19825, 19927, 20158, 20212, 20428, 20458, 20583, 20788, 21276, 21321, 21493, 21817, 21895, 22262, 22303, 22344, 22879, 23371, 24268, 24337, 24979 (all at n=5K)

20479 (4917)

17536 (4845)

13165 (4713)

14980 (4579)

19751 (4554)

20747 (4471)

19780 (4400)

11971 (4360)

24421 (4047)

21731 (3999)

k = 81, 1024, 2500, 5329, 8281, 12996, 17424, and 24025 proven composite by condition 1.

k = 3240 and 5290 proven composite by condition 2.

41

8

3, 7

none - proven

7 (153)

5 (10)

1 (3)

6 (2)

2 (2)

4 (1)

3 (1)

42

15137

5, 43, 353

603, 1049, 1600, 2538, 4299, 4903, 5118, 5978, 6836, 6964, 6971, 7309, 8297, 8341, 9029, 9201, 9633, 9848, 11267, 11781, 11911, 11996, 12125, 12127, 12213, 12598, 13288, 13347, 14884 (k = 1600, 6971 and 14884 at n=8K, other k at n=200K)

7051 (188034)

5417 (179220)

13898 (152983)

1633 (128734)

13757 (126934)

7913 (108747)

15024 (104613)

8453 (89184)

7658 (79316)

10923 (61071)

43

21

2, 11

13 (50K)

4 (279)

12 (203)

17 (79)

3 (24)

1 (5)

19 (4)

15 (4)

7 (4)

11 (2)

10 (2)

44

4

3, 5

none - proven

1 (5)

2 (4)

3 (1)

45

93

2, 23

none - proven

24 (153355)

53 (582)

70 (167)

29 (146)

76 (102)

85 (82)

91 (50)

77 (26)

1 (19)

33 (11)

46

928

3, 7, 103

281, 436, 800 (k = 800 at n=500K, other k at n=28K)

870 (51699)

86 (26325)

93 (24162)

561 (5011)

576 (3659)

100 (2955)

386 (2425)

338 (1478)

597 (950)

121 (935)

47

5

2, 3

none - proven

4 (1555)

1 (127)

2 (4)

3 (2)

48

3226

5, 7, 461

313, 384, 708, 909, 916, 1093, 1457, 1686, 1877, 1896, 1898, 2071, 2148, 2172, 2402, 2589, 2682, 2927, 2939, 3044, 3067 (all at n=200K)

2157 (169491)

2549 (169453)

1478 (167541)

2822 (129611)

2379 (116204)

118 (107422)

692 (103056)

1842 (87175)

953 (81493)

2582 (75696)

49

81

2, 5

All k = m^2 for all n;

factors to:

(m*7^n - 1) *

(m*7^n + 1)

none - proven

79 (212)

44 (122)

69 (42)

30 (24)

59 (16)

53 (15)

70 (14)

24 (14)

31 (9)

74 (6)

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

50

16

3, 17

none - proven

14 (66)

13 (19)

5 (12)

11 (6)

6 (6)

1 (3)

8 (2)

2 (2)

15 (1)

12 (1)

51

25

2, 13

none - proven

1 (4229)

23 (96)

3 (8)

12 (4)

14 (3)

4 (3)

22 (2)

19 (2)

18 (2)

15 (2)

52

25015

3, 7, 53, 379

(Condition 1):

All k where k = m^2

and m = = 23 or 30 mod 53:

for even n let k = m^2

and let n = 2*q; factors to:

(m*52^q - 1) *

(m*52^q + 1)

odd n:

factor of 53

(Condition 2):

All k where k = 13*m^2

and m = = 7 or 46 mod 53:

even n:

factor of 53

for odd n let k = 13*m^2

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

[m*2^(2q-1)*13^q - 1] *

[m*2^(2q-1)*13^q + 1]

82, 349, 372, 476, 478, 657, 796, 902, 1167, 1234, 1271, 1534, 1589, 1651, 1669, 1801, 1881, 1909, 2035, 2113, 2364, 2437, 2492, 2557, 2643, 2722, 2725, 2769, 3022, 3128, 3199, 3229, 3418, 3559, 3607, 3656, 3764, 3788, 3847, 3870, 4043, 4117, 4239, 4294, 4329, 4366, 4597, 4665, 4754, 4975, 4981, 5037, 5107, 5142, 5158, 5246, 5541, 5575, 5672, 5836, 5882, 6193, 6256, 6308, 6394, 6442, 6493, 6568, 6697, 6835, 6873, 6962, 6981, 6997, 7386, 7399, 7594, 7633, 8163, 8389, 8422, 8488, 8587, 8693, 8744, 8932, 8958, 9055, 9148, 9187, 9223, 9382, 9421, 9624, 9647, 9667, 9682, 9753, 9769, 9799, 9802, 9907, 9967, 10069, 10129, 10173, 10243, 10429, 10462, 10546, 10919, 10996, 11161, 11164, 11299, 11355, 11371, 11394, 11401, 11500, 11767, 11826, 11827, 11854, 12064, 12133, 12304, 12352, 12401, 12423, 12454, 12668, 12688, 12719, 12827, 12931, 13045, 13196, 13198, 13264, 13306, 13357, 13551, 13687, 14309, 14453, 14584, 14647, 14682, 14698, 14786, 14833, 14968, 15010, 15109, 15212, 15265, 15316, 15370, 15574, 15688, 15928, 15937, 16007, 16039, 16087, 16111, 16216, 16293, 16308, 16729, 16748, 16884, 16906, 17197, 17224, 17277, 17311, 17423, 17438, 17734, 17754, 17882, 17989, 18604, 18670, 18757, 18761, 18787, 18871, 18883, 18899, 19026, 19028, 19079, 19102, 19163, 19363, 19556, 19609, 19678, 19821, 19876, 19982, 20088, 20139, 20395, 20616, 20821, 20881, 20883, 20983, 21016, 21148, 21151, 21316, 21413, 21464, 21526, 21537, 21757, 21784, 21796, 21804, 21859, 21866, 21898, 22096, 22146, 22180, 22308, 22312, 22383, 22447, 22471, 22643, 22723, 22738, 22771, 22789, 23215, 23268, 23344, 23377, 23427, 23518, 23531, 23533, 23584, 23692, 23773, 24331, 24403, 24557, 24591, 24911 (all at n=5K)

24244 (4987)

24503 (4983)

1357 (4981)

607 (4949)

7603 (4924)

14998 (4896)

14179 (4797)

6434 (4793)

21572 (4673)

5236 (4447)

k = 529, 900, 5776, 6889, 16641, and 18496 proven composite by condition 1.

k = 637 proven composite by condition 2.

53

13

2, 3

none - proven

12 (71)

10 (71)

2 (44)

7 (11)

1 (11)

8 (8)

11 (6)

9 (3)

5 (2)

6 (1)

54

21

5, 11

(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]

none - proven

20 (8)

19 (6)

10 (4)

17 (3)

1 (3)

14 (2)

7 (2)

3 (2)

18 (1)

16 (1)

k = 4 and 9 proven composite by condition 1.

k = 6 proven composite by condition 2.

55

13

2, 7

none - proven

3 (76)

1 (17)

11 (8)

9 (3)

7 (2)

6 (2)

12 (1)

10 (1)

8 (1)

5 (1)

56

20

3, 19

none - proven

14 (26)

10 (23)

1 (7)

18 (4)

17 (4)

7 (3)

11 (2)

8 (2)

5 (2)

2 (2)

57

144

5, 13, 29

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

87 (242)

54 (157)

100 (109)

59 (83)

115 (34)

124 (31)

88 (27)

63 (22)

139 (20)

38 (20)

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

58

547

3, 7, 163

71, 130, 169, 178, 319, 456, 493, 499 (k = 71 and 456 at n=100K, other k at n=14K)

382 (7188)

400 (5245)

421 (4526)

176 (2854)

473 (1641)

487 (1412)

312 (1079)

334 (724)

53 (645)

457 (492)

59

4

3, 5

none - proven

3 (8)

1 (3)

2 (2)

60

20558

13, 61, 277

(Condition 1):

All k where k = m^2

and m = = 11 or 50 mod 61:

for even n let k = m^2

and let n = 2*q; factors to:

(m*60^q - 1) *

(m*60^q + 1)

odd n:

factor of 61

(Condition 2):

All k where k = 15*m^2

and m = = 22 or 39 mod 61:

even n:

factor of 61

for odd n let k = 15*m^2

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

[m*2^(2q-1)*15^q - 1] *

[m*2^(2q-1)*15^q + 1]

36, 1770, 4708, 5317, 5611, 6101, 6162, 6274, 7060, 7870, 8722, 9212, 9454, 9881, 10249, 11101, 12061, 12072, 12098, 12479, 12996, 13297, 13480, 14275, 14851, 15800, 16167, 17185, 17620, 18055, 18965, 18972, 19336, 19394, 19397 (k = 16167 and 18055 at n=8K, other k at n=100K)

1024 (90701)

12121 (84208)

15227 (80625)

15185 (79350)

8649 (79159)

20131 (71977)

19457 (68854)

16333 (61172)

18776 (60164)

1486 (58932)

k = 121, 2500, 5184, 14641, and 17689 proven composite by condition 1.

k = 7260 proven composite by condition 2.

61

125

2, 31

37, 53, 100 (all at n=10K)

13 (4134)

77 (3080)

10 (1552)

41 (755)

42 (174)

22 (117)

57 (89)

109 (86)

103 (78)

93 (60)

62

8

3, 7

none - proven

3 (59)

4 (9)

1 (3)

6 (2)

5 (2)

2 (2)

7 (1)

63

857

2, 5, 397

93, 129, 139, 211, 231, 237, 251, 281, 291, 333, 417, 457, 471, 473, 491, 493, 497, 513, 587, 599, 633, 669, 677, 679, 691, 733, 771, 817, 819, 831 (all at n=2K)

65 (1883)

853 (1849)

37 (1615)

64 (1483)

177 (1423)

372 (1320)

821 (1225)

687 (1154)

695 (1144)

271 (1058)

64

14

5, 13

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

11 (9)

12 (6)

5 (2)

13 (1)

10 (1)

7 (1)

6 (1)

3 (1)

2 (1)

k = 1, 4, 8, and 9 proven composite by full algebraic factors.

65

10

3, 11

none - proven

1 (19)

8 (10)

4 (9)

2 (4)

5 (2)

9 (1)

7 (1)

6 (1)

3 (1)

66

63717671

7, 67, 613, 4423

681, 1056, 1205, 1575, 1669, 1944, 2182, 2916, 2949, 3014, 3083, 3148, 3221, 3526, 3684, 3911, 3946, 4423, 5329, 5361, 5897, 5898, 5959, 5972, 6096, 6189, 6263, 6451, 6768, 6796, 7168, 7237, 7357, 7572, 7614, 7927, 8156, 8173, 8348, 8432, 8510, 8825, 8866, 9017, 9111, 9406, 9409, 9781, 9801, 9906, 9998 (for k <= 10K) (all at n=1K)

7578 (988)

1252 (956)

2746 (918)

5248 (916)

5476 (873)

5929 (795)

6699 (790)

8843 (780)

5435 (762)

2946 (748)

67

33

2, 17

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*67^q - 1) *

(m*67^q + 1)

odd n:

factor of 17

none - proven

25 (2829)

2 (768)

23 (42)

21 (27)

1 (19)

31 (10)

19 (8)

18 (7)

13 (7)

11 (6)

k = 16 proven composite by partial algebraic factors.

68

22

3, 23

none - proven

7 (25395)

5 (13574)

11 (198)

8 (62)

10 (53)

3 (10)

1 (5)

14 (4)

2 (4)

9 (3)

69

6

3, 5

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*69^q - 1) *

(m*69^q + 1)

odd n:

factor of 5

none - proven

5 (4)

1 (3)

3 (1)

2 (1)

k = 4 proven composite by partial algebraic factors.

70

853

13, 29, 71

811 (50K)

729 (28625)

376 (6484)

496 (4934)

434 (3820)

489 (2096)

278 (1320)

550 (764)

31 (545)

174 (441)

778 (356)

71

5

2, 3

none - proven

2 (52)

1 (3)

3 (2)

4 (1)

72

293

5, 17, 73

none - proven

4 (1119849)

79 (28009)

291 (26322)

116 (13887)

118 (4599)

67 (4308)

197 (3256)

24 (2648)

11 (2445)

18 (1494)

73

112

5, 13, 37

(Condition 1):

All k where k = m^2

and m = = 6 or 31 mod 37:

for even n let k = m^2

and let n = 2*q; factors to:

(m*73^q - 1) *

(m*73^q + 1)

odd n:

factor of 37

(Condition 2):

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*73^q - 1) *

(m*73^q + 1)

odd n:

factor of 2

none - proven

79 (9339)

101 (2146)

105 (102)

48 (73)

54 (63)

42 (50)

26 (50)

97 (47)

61 (39)

89 (32)

k = 36 proven composite by condition 1.

k = 9 and 25 proven composite by condition 2.

74

4

3, 5

none - proven

2 (132)

1 (5)

3 (2)

75

37

2, 19

none - proven

35 (1844)

16 (119)

18 (54)

30 (41)

3 (16)

22 (15)

5 (9)

17 (5)

4 (5)

23 (4)

76

34

7, 11

none - proven

1 (41)

27 (40)

20 (22)

25 (11)

15 (11)

30 (7)

21 (4)

19 (4)

13 (4)

10 (4)

77

13

2, 3

none - proven

2 (14)

1 (3)

12 (2)

11 (2)

8 (2)

5 (2)

3 (2)

10 (1)

9 (1)

7 (1)

78

90059

5, 79, 1217

274, 302, 631, 1816, 2292, 2381, 3872, 3949, 4344, 4383, 4489, 4937, 5057, 5766, 5782, 6077, 6436, 7032, 7800, 8469, 8499, 8649, 8758, 10263, 10924, 10928, 10942, 11044, 11936, 12167, 12187, 12244, 12286, 12332, 12622, 13212, 13287, 13668, 13824, 14059, 14456, 14526, 14932, 15722, 15799, 16451, 16688, 17029, 17039, 17221, 17271, 17732, 17886, 18013, 18663, 19614, 19846, 19909, 19986, 20027, 20182, 20462, 20879, 21197, 21631, 21961, 23052, 23079, 23801, 23899, 24214, 24949, 25061, 25532, 25901, 26377, 26385, 26804, 27021, 27096, 27175, 27256, 27399, 27439, 27842, 29073, 29389, 29668, 29863, 30444, 31046, 31053, 31742, 31836, 31917, 31994, 32705, 33298, 33412, 33671, 33888, 33892, 34728, 35179, 35568, 36233, 36344, 36609, 37024, 38354, 38438, 38711, 38886, 39173, 39901, 40131, 40239, 40289, 40437, 40998, 41079, 41316, 41711, 41748, 42106, 42337, 42896, 43331, 43842, 43886, 44038, 44374, 44634, 44871, 45214, 45221, 45466, 46012, 46187, 46593, 46922, 47004, 47562, 47573, 47636, 47657, 47986, 48004, 48112, 48371, 48973, 48979, 49386, 49611, 49988, 51430, 52042, 52929, 53719, 53761, 54188, 54936, 55245, 55491, 55617, 56563, 56721, 56757, 56904, 57234, 57317, 57611, 57786, 57842, 58402, 58455, 58696, 58854, 59093, 59536, 59774, 60187, 60919, 60978, 61762, 61783, 61937, 62481, 62646, 62854, 63043, 63281, 63351, 64309, 64384, 64744, 65157, 65814, 65885, 66102, 66249, 66991, 67386, 67588, 67593, 67706, 67880, 68027, 68573, 68804, 69630, 69914, 71254, 71338, 72003, 72916, 72997, 73706, 73708, 73734, 73787, 74757, 74823, 75307, 75482, 75857, 75888, 76056, 76392, 76781, 77057, 77594, 78135, 78604, 78835, 78959, 79630, 79633, 79674, 80421, 80725, 80788, 80976, 81208, 81369, 83186, 83739, 84484, 85218, 85506, 85886, 86137, 86164, 86329, 86353, 86446, 86692, 88718, 88817, 88866, 89314, 89538, 89664, 89846 (k = 1 mod 7 and k = 1 mod 11 at n=1K, other k at n=100K)

3633 (94500)

68571 (91386)

51476 (88677)

78053 (84433)

58412 (83824)

45661 (73022)

11412 (72798)

72638 (70230)

23462 (69162)

23543 (62677)

79

9

2, 5

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*79^q - 1) *

(m*79^q + 1)

odd n:

factor of 5

none - proven

1 (5)

7 (4)

3 (4)

6 (3)

8 (1)

5 (1)

2 (1)

k = 4 proven composite by partial algebraic factors.

80

253

3, 37, 173

10, 31, 214 (all at n=400K)

170 (148256)

106 (16237)

154 (9753)

46 (5337)

232 (2997)

157 (2613)

169 (1959)

45 (1156)

218 (776)

244 (653)

81

74

7, 13, 73

All k = m^2 for all n;

factors to:

(m*9^n - 1) *

(m*9^n + 1)

none - proven

53 (268)

42 (99)

23 (68)

18 (15)

35 (14)

30 (12)

71 (4)

60 (4)

40 (4)

24 (4)

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

82

22326

5, 83, 269

118, 133, 290, 331, 334, 439, 625, 649, 667, 748, 757, 763, 829, 878, 883, 898, 997, 1163, 1252, 1279, 1327, 1348, 1351, 1531, 1741, 1827, 1936, 1991, 2050, 2157, 2263, 2278, 2419, 2431, 2539, 2543, 2588, 2635, 2668, 2797, 2836, 2896, 2929, 2971, 2974, 3079, 3121, 3156, 3293, 3319, 3436, 3653, 3796, 3817, 4068, 4078, 4083, 4118, 4372, 4399, 4447, 4481, 4483, 4780, 4801, 4867, 4898, 4972, 5053, 5182, 5230, 5311, 5329, 5401, 5560, 5562, 5713, 5893, 5899, 5975, 6028, 6122, 6124, 6143, 6178, 6186, 6226, 6296, 6343, 6418, 6427, 6571, 6631, 6925, 6994, 7054, 7056, 7303, 7386, 7388, 7396, 7474, 7615, 7723, 7801, 7813, 7822, 7884, 7892, 7969, 8065, 8314, 8368, 8384, 8499, 8629, 8761, 8830, 8878, 8891, 8941, 9124, 9166, 9304, 9409, 9461, 9712, 9739, 9967, 9988, 10000, 10036, 10075, 10147, 10162, 10448, 10542, 10891, 10957, 11056, 11086, 11119, 11123, 11271, 11372, 11485, 11533, 11553, 11665, 11728, 11827, 11884, 11929, 12079, 12169, 12202, 12211, 12283, 12547, 12562, 12587, 12791, 13126, 13141, 13358, 13531, 13613, 13768, 13779, 13792, 13862, 13891, 14095, 14109, 14161, 14188, 14242, 14257, 14275, 14349, 14441, 14524, 14531, 14563, 14614, 14687, 14855, 14939, 14941, 14986, 15046, 15136, 15271, 15343, 15349, 15403, 15493, 15508, 15634, 15679, 15682, 15852, 15997, 16024, 16103, 16131, 16242, 16312, 16534, 16633, 16753, 16756, 16767, 16954, 17011, 17401, 17512, 17518, 17761, 17803, 17833, 17878, 18058, 18061, 18431, 18448, 18514, 18538, 18550, 18757, 19093, 19237, 19309, 19372, 19414, 19444, 19519, 19672, 19678, 19930, 19946, 20002, 20050, 20113, 20218, 20251, 20413, 20491, 20578, 20581, 20708, 20773, 20980, 21052, 21088, 21215, 21282, 21334, 21382, 21398, 21433, 21449, 21453, 21454, 21466, 21514, 21541, 21631, 21683, 21762, 21862, 21871, 21913, 22012, 22132, 22162, 22243, 22245 (k = 1 mod 3 at n=1K, other k at n=100K)

15978 (99999)

21429 (96772)

18989 (96049)

17592 (83837)

22233 (75716)

12912 (74869)

5811 (72615)

16091 (65850)

18576 (64927)

4482 (63245)

83

5

2, 3

none - proven

2 (8)

1 (5)

3 (2)

4 (1)

84

16

5, 17

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*84^q - 1) *

(m*84^q + 1)

odd n:

factor of 5

none - proven

1 (17)

14 (8)

11 (7)

8 (4)

12 (3)

15 (1)

13 (1)

10 (1)

7 (1)

6 (1)

k = 4 and 9 proven composite by partial algebraic factors.

85

173

2, 43

61 (15K)

169 (6939)

64 (1253)

105 (403)

112 (394)

97 (287)

109 (230)

16 (171)

27 (160)

93 (90)

145 (77)

86

28

3, 29

none - proven

23 (112)

14 (38)

18 (26)

27 (14)

1 (11)

2 (10)

25 (9)

11 (8)

22 (5)

19 (5)

87

21

2, 11

none - proven

19 (372)

9 (91)

16 (17)

18 (15)

5 (15)

13 (11)

11 (10)

1 (7)

7 (6)

12 (5)

88

571

3, 7, 13, 19

k = 400:

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

(20*88^q - 1) *

(20*88^q + 1)

odd n:

covering set 3, 7, 13

46, 94, 277, 508 (all at n=10K)

464 (20648)

444 (19708)

544 (8904)

380 (8712)

79 (7665)

477 (5816)

212 (5511)

179 (4545)

346 (2969)

68 (2477)

89

4

3, 5

none - proven

2 (60)

3 (5)

1 (3)

90

27

7, 13

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*90^q - 1) *

(m*90^q + 1)

odd n:

factor of 13

none - proven

6 (20)

11 (10)

10 (10)

13 (6)

15 (5)

12 (4)

7 (4)

24 (3)

1 (3)

20 (2)

k = 25 proven composite by partial algebraic factors.

91

45

2, 23

none - proven

27 (5048)

1 (4421)

37 (159)

15 (14)

43 (6)

39 (6)

31 (6)

24 (5)

20 (4)

36 (3)

92

32

3, 31

none - proven

1 (439)

29 (272)

28 (99)

13 (35)

14 (32)

18 (26)

22 (25)

20 (6)

6 (6)

17 (4)

93

189

2, 47

33, 69, 109, 113, 125, 149, 177 (all at n=8K)

97 (1179)

29 (496)

92 (476)

46 (434)

121 (271)

141 (262)

101 (142)

122 (126)

85 (86)

166 (66)

94

39

5, 19

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*94^q - 1) *

(m*94^q + 1)

odd n:

factor of 5

29 (1M)

16 (21951)

37 (254)

13 (163)

14 (154)

7 (95)

34 (54)

25 (41)

24 (12)

26 (9)

36 (7)

k = 4 and 9 proven composite by partial algebraic factors.

95

5

2, 3

none - proven

1 (7)

3 (2)

2 (2)

4 (1)

96

38995

7, 67, 97, 1303

(Condition 1):

All k where k = m^2

and m = = 22 or 75 mod 97:

for even n let k = m^2

and let n = 2*q; factors to:

(m*96^q - 1) *

(m*96^q + 1)

odd n:

factor of 97

(Condition 2):

All k where k = 6*m^2

and m = = 9 or 88 mod 97:

even n:

factor of 97

for odd n let k = 6*m^2

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

[m*2^(5q-1)*3^q - 1] *

[m*2^(5q-1)*3^q + 1]

431, 591, 701, 831, 872, 956, 1006, 1126, 1648, 1681, 1810, 2036, 2386, 2424, 2878, 3001, 3431, 3461, 3671, 3856, 3881, 3956, 3996, 4261, 4351, 4366, 4406, 4451, 4461, 5046, 5836, 5918, 6031, 6261, 6481, 6586, 6670, 6786, 7091, 7116, 7121, 7131, 7249, 7274, 7461, 7801, 8016, 8202, 8291, 8546, 8816, 9022, 9131, 9156, 9326, 9441, 9463, 9476, 9677, 9681, 9921, 10036, 10204, 10375, 10453, 10551, 10651, 10721, 11056, 11156, 11196, 11458, 11553, 11766, 11831, 12676, 12901, 13216, 13231, 13288, 13571, 14011, 14061, 14276, 14517, 14551, 14646, 15341, 15461, 15573, 15596, 16176, 16306, 16392, 16586, 16641, 16645, 17116, 17421, 17636, 17653, 17792, 18311, 19136, 19191, 19246, 19486, 19681, 20091, 20396, 20464, 20502, 20936, 21488, 21776, 22541, 22811, 22846, 22931, 23010, 23161, 23271, 23301, 23570, 23766, 24076, 24216, 24386, 24506, 24831, 24916, 24929, 25306, 25706, 25966, 26038, 26161, 26183, 26571, 26772, 26801, 26846, 27045, 27106, 27126, 27450, 27646, 27700, 27741, 28365, 28558, 28774, 28776, 28921, 29093, 29196, 29561, 29681, 30086, 30120, 30151, 30421, 30581, 30662, 31021, 31136, 31936, 32205, 32881, 33099, 33141, 33391, 33406, 33501, 33621, 33701, 33711, 33951, 33986, 34116, 34236, 34436, 34531, 34921, 35016, 35113, 35271, 35406, 35446, 35781, 35966, 36158, 36551, 36945, 36981, 37031, 37036, 37166, 37222, 37471, 37991, 38156, 38301, 38316, 38986 (k = 1 mod 5 and k = 1 mod 19 at n=1K, other k at n=100K)

3769 (92879)

28907 (89447)

13528 (86114)

19882 (82073)

37155 (76817)

9160 (71178)

5179 (66965)

32960 (60312)

7565 (59052)

4754 (56909)

k = 484, 5625, 14161, and 29584 proven composite by condition 1.

k = 486 proven composite by condition 2.

97

43

3, 5, 7, 37, 139

22 (35.8K)

8 (192335)

16 (1627)

4 (621)

28 (184)

1 (17)

34 (16)

32 (9)

27 (8)

37 (5)

31 (5)

98

10

3, 11

none - proven

1 (13)

5 (10)

7 (3)

4 (3)

8 (2)

2 (2)

9 (1)

6 (1)

3 (1)

99

9

2, 5

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*99^q - 1) *

(m*99^q + 1)

odd n:

factor of 5

none - proven

5 (135)

3 (4)

1 (3)

7 (2)

8 (1)

6 (1)

2 (1)

k = 4 proven composite by partial algebraic factors.

100

211

7, 13, 37

All k = m^2 for all n;

factors to:

(m*10^n - 1) *

(m*10^n + 1)

none - proven

74 (44709)

133 (5496)

102 (209)

193 (155)

203 (133)

95 (96)

109 (68)

55 (56)

98 (45)

37 (36)

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

101

13

2, 3

none - proven

5 (350)

8 (112)

2 (42)

11 (24)

12 (11)

4 (3)

1 (3)

6 (2)

10 (1)

9 (1)

102

1635

7, 19, 79

191, 207, 1082, 1369 (all at n=500K)

1451 (188973)

1208 (178632)

653 (117255)

1607 (82644)

254 (58908)

1527 (49462)

1037 (43460)

32 (43302)

1296 (37715)

142 (22025)

103

25

2, 13

none - proven

19 (820)

22 (442)

23 (216)

14 (189)

16 (57)

11 (54)

24 (32)

15 (32)

1 (19)

20 (5)

104

4

3, 5

none - proven

1 (97)

2 (68)

3 (1)

105

297

2, 37, 149

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

73, 137 (both at n=8K)

148 (3645)

265 (1666)

162 (294)

255 (222)

154 (139)

145 (119)

80 (91)

68 (56)

66 (47)

223 (21)

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

106

13624

3, 19, 199

64, 81, 163, 332, 391, 400, 511, 526, 643, 676, 841, 862, 897, 1024, 1223, 1283, 1417, 1546, 1597, 1713, 1869, 2116, 2248, 2389, 2458, 2605, 2623, 2674, 2743, 2780, 2781, 2965, 3241, 3277, 3336, 3425, 3427, 3478, 3481, 3617, 3622, 3646, 3655, 3746, 3883, 4045, 4067, 4096, 4153, 4177, 4219, 4336, 4339, 4416, 4628, 4666, 4696, 4713, 4722, 5135, 5283, 5395, 5468, 5623, 5692, 5707, 5752, 5776, 5872, 5878, 5971, 5992, 6094, 6100, 6220, 6376, 6421, 6547, 6613, 6716, 6736, 6784, 6832, 6955, 7069, 7156, 7202, 7246, 7273, 7297, 7331, 7336, 7345, 7398, 7496, 7540, 7561, 7744, 7894, 7906, 8023, 8181, 8266, 8323, 8371, 8386, 8428, 8521, 8572, 8586, 8637, 8779, 8788, 8861, 8950, 8956, 8962, 8975, 9031, 9096, 9190, 9294, 9415, 9469, 9634, 9736, 9787, 9796, 9808, 9859, 9877, 9973, 10033, 10072, 10117, 10166, 10186, 10271, 10273, 10446, 10627, 10646, 10651, 10660, 10699, 10876, 10894, 11173, 11278, 11299, 11426, 11506, 11833, 11884, 11901, 12066, 12090, 12145, 12352, 12490, 12627, 12851, 12856, 12916, 12970, 12991, 13162, 13174, 13366, 13374, 13378, 13387, 13497, 13516, 13528, 13543 (all at n=2K)

913 (1991)

7771 (1952)

13023 (1951)

8561 (1927)

13567 (1850)

12361 (1830)

12910 (1817)

6181 (1800)

2719 (1769)

11639 (1746)

107

5

2, 3

none - proven

2 (21910)

3 (4900)

4 (251)

1 (17)

108

13406

7, 13, 61, 109

(Condition 1):

All k where k = m^2

and m = = 33 or 76 mod 109:

for even n let k = m^2

and let n = 2*q; factors to:

(m*108^q - 1) *

(m*108^q + 1)

odd n:

factor of 109

(Condition 2):

All k where k = 3*m^2

and m = = 20 or 89 mod 109:

even n:

factor of 109

for odd n let k = 3*m^2

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

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

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

137, 411, 437, 873, 1634, 1769, 1782, 1961, 2508, 2617, 2962, 2963, 3002, 3029, 3474, 3499, 3596, 3646, 4007, 4066, 4084, 4121, 4184, 4328, 4468, 4499, 4744, 4904, 5015, 5142, 5212, 5351, 5625, 5821, 5892, 5923, 5994, 6212, 6284, 6432, 6528, 6570, 6614, 6866, 7107, 7211, 7302, 7304, 7419, 7848, 8037, 8144, 8374, 8383, 8503, 8524, 8638, 8986, 9346, 9852, 10052, 10129, 10136, 10245, 10699, 10926, 11089, 11164, 11278, 11619, 11881, 11918, 12262, 12861, 12863, 13162, 13291, 13297 (k = 5351, 6528, and 13162 at n=6K, other k at n=100K)

10322 (88080)

1999 (85188)

7557 (84180)

11882 (81547)

3439 (79524)

4686 (79010)

1159 (77107)

3573 (76352)

1465 (75209)

2148 (75018)

k = 1089 and 5776 proven composite by condition 1.

k = 1200 proven composite by condition 2.

109

9

2, 5

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*109^q - 1) *

(m*109^q + 1)

odd n:

factor of 5

none - proven

8 (19)

1 (17)

5 (2)

2 (2)

7 (1)

6 (1)

3 (1)

k = 4 proven composite by partial algebraic factors.

110

38

3, 37

All k where k = m^2

and m = = 6 or 31 mod 37:

for even n let k = m^2

and let n = 2*q; factors to:

(m*110^q - 1) *

(m*110^q + 1)

odd n:

factor of 37

none - proven

23 (78120)

17 (2598)

37 (1689)

9 (77)

11 (42)

10 (17)

2 (16)

31 (9)

5 (6)

22 (5)

k = 36 proven composite by partial algebraic factors.

111

13

2, 7

none - proven

2 (24)

7 (6)

6 (4)

1 (3)

12 (2)

11 (2)

3 (2)

10 (1)

9 (1)

8 (1)

112

1357

5, 13, 113

All k where k = m^2

and m = = 15 or 98 mod 113:

for even n let k = m^2

and let n = 2*q; factors to:

(m*112^q - 1) *

(m*112^q + 1)

odd n:

factor of 113

31, 79, 310, 340, 421, 424, 451, 529, 703, 940, 1018, 1051, 1204 (all at n=7.5K)

948 (173968)

1268 (50536)

758 (35878)

1353 (7751)

187 (7524)

498 (6038)

9 (5717)

1024 (5681)

619 (5441)

981 (2858)

k = 225 proven composite by partial algebraic factors.

113

20

3, 19

none - proven

14 (308)

1 (23)

7 (15)

19 (11)

5 (8)

16 (5)

3 (5)

12 (3)

4 (3)

18 (2)

114

24

5, 23

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*114^q - 1) *

(m*114^q + 1)

odd n:

factor of 5

none - proven

3 (63)

1 (29)

11 (27)

18 (21)

22 (20)

20 (3)

19 (2)

17 (2)

14 (2)

10 (2)

k = 4 and 9 proven composite by partial algebraic factors.

115

57

2, 29

13, 43 (both at n=8K)

45 (5227)

4 (4223)

51 (2736)

23 (1116)

53 (165)

21 (127)

35 (50)

15 (38)

39 (28)

32 (28)

116

14

3, 13

none - proven

9 (249)

5 (156)

11 (118)

1 (59)

2 (32)

13 (15)

10 (11)

12 (2)

8 (2)

7 (1)

117

149

2, 5, 37

5, 17, 33, 141 (all at n=8K)

83 (442)

59 (352)

19 (336)

110 (232)

143 (222)

41 (209)

87 (177)

129 (165)

118 (136)

92 (129)

118

50

7, 17

43 (37K)

27 (860)

29 (599)

18 (393)

6 (210)

22 (191)

8 (85)

19 (72)

7 (52)

42 (30)

37 (27)

119

4

3, 5

none - proven

2 (28)

3 (6)

1 (3)

120

166616308

11, 13, 1117, 14281

384, 386, 419, 483, 551, 672, 824, 846, 890, 901, 991, 1024, 1077, 1095, 1132, 1134, 1255, 1309, 1385, 1394, 1693, 1797, 1921, 2036, 2133, 2177, 2258, 2354, 2386, 2410, 2452, 2650, 2696, 2716, 3004, 3025, 3123, 3178, 3189, 3214, 3290, 3343, 3347, 3400, 3407, 3433, 3596, 3786, 3994, 4003, 4082, 4320, 4399, 4423, 4460, 4500, 4577, 4676, 4685, 4819, 4830, 4839, 4936, 5105, 5125, 5255, 5378, 5630, 5686, 5730, 6112, 6241, 6332, 6357, 6425, 6581, 6676, 6678, 6755, 6821, 6852, 6951, 6982, 6997, 7008, 7413, 7470, 7523, 7545, 7549, 7789, 7803, 7820, 7910, 7985, 8100, 8205, 8464, 8647, 8810, 8812, 8869, 8922, 8964, 8966, 8997, 9010, 9019, 9057, 9070, 9395, 9564, 9626, 9712, 9889, 9921, 9954, 9993 (for k <= 10K) (all at n=1K)

8063 (997)

6434 (976)

2980 (958)

5180 (938)

164 (878)

4234 (876)

7085 (843)

4390 (833)

9354 (829)

2726 (822)

121

100

3, 7, 37

All k = m^2 for all n;

factors to:

(m*11^n - 1) *

(m*11^n + 1)

none - proven

62 (13101)

79 (4545)

43 (68)

7 (60)

30 (24)

60 (12)

87 (11)

39 (11)

57 (10)

50 (10)

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

122

14

3, 5, 13

none - proven

13 (43)

8 (26)

11 (10)

2 (6)

12 (5)

1 (5)

10 (3)

6 (2)

5 (2)

3 (2)

123

13

2, 5, 17

11 (8K)

1 (43)

3 (8)

2 (8)

12 (7)

6 (7)

9 (5)

7 (2)

10 (1)

8 (1)

5 (1)

124

92881

3, 5, 7, 5167

(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*124^q - 1) *

(m*124^q + 1)

odd n:

factor of 5

(Condition 2):

All k where k = 31*m^2

and m = = 1 or 4 mod 5:

even n:

factor of 5

for odd n let k = 31*m^2

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

[m*2^(2q-1)*31^q - 1] *

[m*2^(2q-1)*31^q + 1]

101, 136, 146, 175, 179, 199, 204, 236, 259, 271, 301, 328, 364, 389, 434, 441, 459, 469, 561, 586, 589, 599, 604, 614, 616, 631, 661, 741, 766, 806, 844, 894, 901, 922, 931, 951, 971, 974, 1013, 1016, 1019, 1021, 1039, 1043, 1046, 1061, 1081, 1114, 1123, 1149, 1156, 1186, 1229, 1231, 1237, 1246, 1249, 1269, 1288, 1336, 1375, 1376, 1384, 1399, 1461, 1496, 1498, 1499, 1509, 1511, 1519, 1522, 1542, 1636, 1654, 1664, 1711, 1719, 1724, 1731, 1741, 1743, 1754, 1766, 1779, 1783, 1784, 1789, 1814, 1824, 1834, 1861, 1904, 1924, 1926, 1931, 1941, 1954, 1969, 1989, 2029, 2041, 2095, 2101, 2109, 2124, 2131, 2161, 2166, 2191, 2194, 2212, 2296, 2306, 2307, 2344, 2364, 2366, 2377, 2416, 2419, 2436, 2479, 2491, 2497, 2529, 2539, 2559, 2572, 2576, 2616, 2656, 2661, 2664, 2666, 2680, 2686, 2731, 2761, 2789, 2804, 2830, 2854, 2864, 2920, 2931, 2971, 2994, 3024, 3034, 3054, 3067, 3076, 3079, 3081, 3096, 3154, 3196, 3214, 3229, 3247, 3261, 3286, 3294, 3316, 3319, 3324, 3329, 3346, 3382, 3421, 3439, 3579, 3604, 3606, 3646, 3649, 3654, 3679, 3704, 3716, 3730, 3734, 3739, 3752, 3771, 3779, 3786, 3789, 3809, 3821, 3829, 3839, 3866, 3942, 3949, 3964, 3986, 4006, 4015, 4039, 4054, 4066, 4084, 4089, 4091, 4094, 4096, 4129, 4134, 4153, 4207, 4229, 4231, 4234, 4236, 4311, 4319, 4331, 4375, 4376, 4384, 4424, 4429, 4476, 4486, 4506, 4512, 4526, 4546, 4554, 4609, 4646, 4651, 4684, 4714, 4716, 4771, 4786, 4796, 4801, 4811, 4816, 4831, 4854, 4879, 4885, 4909, 4911, 4946, 4961, 4976, 4997, 5009, 5020, 5026, 5032, 5049, 5101, 5116, 5149, 5152, 5164, 5186, 5209, 5224, 5226, 5246, 5269, 5274, 5283, 5314, 5334, 5396, 5404, 5416, 5431, 5459, 5499, 5526, 5539, 5554, 5611, 5626, 5630, 5632, 5679, 5684, 5696, 5699, 5710, 5746, 5751, 5764, 5784, 5830, 5840, 5844, 5911, 5926, 5934, 5946, 5956, 5959, 5974, 5979, 5982, 6000, 6019, 6024, 6049, 6094, 6098, 6106, 6154, 6181, 6184, 6186, 6187, 6189, 6191, 6212, 6214, 6223, 6226, 6246, 6251, 6261, 6309, 6318, 6336, 6361, 6374, 6376, 6381, 6384, 6424, 6434, 6439, 6449, 6466, 6469, 6506, 6514, 6571, 6589, 6625, 6644, 6759, 6799, 6826, 6849, 6856, 6886, 6901, 6919, 6931, 6961, 6971, 6976, 6986, 7006, 7051, 7062, 7066, 7092, 7096, 7104, 7114, 7134, 7144, 7146, 7195, 7221, 7232, 7261, 7274, 7276, 7284, 7301, 7309, 7311, 7329, 7369, 7389, 7396, 7423, 7453, 7456, 7478, 7479, 7494, 7516, 7521, 7522, 7523, 7544, 7551, 7591, 7600, 7616, 7617, 7619, 7674, 7682, 7714, 7739, 7741, 7756, 7762, 7771, 7779, 7801, 7811, 7861, 7884, 7885, 7897, 7909, 7951, 8006, 8041, 8044, 8046, 8111, 8124, 8129, 8137, 8146, 8149, 8161, 8166, 8201, 8203, 8231, 8248, 8249, 8250, 8266, 8286, 8326, 8334, 8339, 8361, 8369, 8383, 8394, 8419, 8429, 8431, 8441, 8454, 8461, 8476, 8479, 8491, 8499, 8524, 8529, 8536, 8551, 8564, 8581, 8606, 8641, 8655, 8674, 8683, 8691, 8719, 8724, 8730, 8779, 8794, 8809, 8811, 8839, 8849, 8854, 8869, 8871, 8934, 8936, 8974, 8979, 8980, 8986, 9001, 9034, 9064, 9069, 9076, 9115, 9136, 9142, 9166, 9172, 9175, 9178, 9199, 9236, 9244, 9247, 9256, 9260, 9264, 9276, 9314, 9334, 9336, 9344, 9349, 9366, 9382, 9401, 9436, 9454, 9459, 9463, 9496, 9516, 9524, 9526, 9551, 9562, 9564, 9571, 9574, 9586, 9634, 9646, 9661, 9728, 9739, 9761, 9799, 9826, 9831, 9844, 9907, 9909, 9931, 9966, 9976 (for k <= 10K) (all at n=1K)

1194 (998)

1611 (989)

659 (986)

3996 (985)

6314 (984)

6101 (983)

4903 (978)

3941 (977)

6011 (975)

6179 (972)

k = 2^2, 3^2, 7^2, 8^2, 12^2, 13^2, 17^2, 18^2 (etc. pattern repeating every 5m) proven composite by condition 1.

k = 31*1^2, 31*4^2, 31*6^2, 31*9^2, 31*11^2, 31*14^2, 31*16^2, 31*19^2 (etc. pattern repeating every 5m) proven composite by condition 2.

125

8

3, 7

All k = m^3 for all n;

factors to:

(m*5^n - 1) *

(m^2*25^n + m*5^n + 1)

none - proven

6 (24)

7 (5)

3 (3)

5 (2)

2 (2)

4 (1)

k = 1 proven composite by full algebraic factors.

126

480821

13, 19, 127, 829

406, 1855, 2707, 2744, 3285, 3566, 3573, 3631, 3721, 4416, 4436, 4596, 5081, 5285, 6026, 6041, 6605, 7075, 7107, 7580, 7876, 8061, 8256, 8323, 8336, 8836, 9166, 9524, 9606, 9651, 9936, 11366, 11475, 11493, 11696, 12013, 12416, 12594, 13006, 13016, 13027, 13302, 13389, 13824, 14270, 14831, 15366, 15596, 15752, 15898, 16636, 16974, 17351, 17436, 17826, 17920, 18001, 18058, 18162, 18430, 18571, 18617, 19686, 19996, 20216, 20575, 20907, 20983, 21306, 21316, 22031, 22389, 22790, 22837, 23390, 23466, 23748, 23903, 24001, 24176, 24706, 25106, 25886, 26326, 26490, 27296, 28791, 28928, 29001, 29012, 29551, 29719 (for k <= 30K) (k = 1 mod 5 at n=1K, other k at n=25K)

8099 (23965)

24832 (23531)

28659 (23470)

20497 (22584)

21342 (22321)

6990 (21006)

26279 (19646)

18638 (17149)

27730 (16804)

29617 (16038)

127

2593

2, 5, 17, 137

13, 17, 25, 27, 33, 35, 79, 83, 91, 113, 121, 139, 159, 179, 191, 231, 233, 235, 236, 237, 239, 250, 251, 264, 279, 288, 293, 333, 353, 361, 367, 379, 443, 451, 459, 471, 473, 511, 513, 517, 523, 531, 537, 551, 553, 557, 561, 597, 599, 604, 617, 631, 639, 649, 659, 679, 699, 715, 725, 731, 733, 737, 739, 747, 751, 755, 763, 773, 778, 783, 797, 809, 838, 848, 863, 871, 895, 919, 937, 939, 950, 953, 964, 982, 997, 999, 1013, 1019, 1025, 1031, 1037, 1039, 1043, 1051, 1106, 1107, 1117, 1119, 1127, 1157, 1173, 1185, 1196, 1199, 1211, 1231, 1232, 1233, 1245, 1253, 1259, 1279, 1288, 1291, 1313, 1327, 1333, 1335, 1337, 1347, 1353, 1359, 1371, 1377, 1401, 1407, 1417, 1421, 1429, 1432, 1439, 1473, 1481, 1491, 1513, 1525, 1539, 1549, 1551, 1573, 1577, 1579, 1589, 1593, 1595, 1597, 1599, 1611, 1612, 1618, 1631, 1639, 1641, 1661, 1677, 1693, 1699, 1709, 1711, 1731, 1732, 1737, 1751, 1771, 1792, 1793, 1803, 1837, 1839, 1903, 1911, 1921, 1928, 1933, 1936, 1939, 1943, 1951, 1957, 1959, 1999, 2013, 2017, 2032, 2039, 2045, 2072, 2073, 2079, 2092, 2097, 2099, 2129, 2155, 2168, 2179, 2191, 2197, 2215, 2231, 2247, 2253, 2273, 2279, 2303, 2313, 2339, 2367, 2377, 2389, 2411, 2427, 2431, 2433, 2479, 2501, 2543, 2548, 2559, 2565, 2573, 2583 (all at n=1K)

667 (1000)

1775 (994)

2497 (989)

2199 (972)

1759 (936)

2015 (910)

343 (904)

1113 (899)

1962 (893)

1543 (872)

128

44

3, 43

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)

none - proven

29 (211192)

23 (2118)

26 (1442)

37 (699)

16 (459)

42 (246)

35 (98)

30 (66)

36 (59)

12 (46)

k = 1 proven composite by full algebraic factors.

129

14

5, 13

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*129^q - 1) *

(m*129^q + 1)

odd n:

factor of 5

none - proven

12 (228)

1 (5)

5 (3)

7 (2)

13 (1)

11 (1)

10 (1)

8 (1)

6 (1)

3 (1)

k = 4 and 9 proven composite by partial algebraic factors.

130

2563

3, 7, 811

64, 247, 253, 254, 302, 597, 739, 799, 877, 918, 961, 1003, 1129, 1159, 1178, 1255, 1258, 1423, 1702, 1754, 1773, 1807, 1849, 2227, 2304, 2311, 2319, 2381, 2479, 2494, 2536 (all at n=2K)

148 (1894)

1555 (1886)

1049 (1881)

2242 (1850)

2326 (1749)

1114 (1724)

523 (1670)

1796 (1650)

557 (1525)

1483 (1490)

131

5

2, 3

none - proven

2 (4)

1 (3)

3 (2)

4 (1)

132

20

7, 19

none - proven

18 (62)

1 (47)

3 (38)

8 (11)

19 (9)

4 (3)

13 (2)

7 (2)

6 (2)

17 (1)

133

17

2, 5, 29

none - proven

1 (13)

11 (5)

2 (4)

12 (3)

9 (3)

7 (3)

4 (3)

13 (2)

5 (2)

16 (1)

134

4

3, 5

none - proven

1 (5)

2 (2)

3 (1)

135

33

2, 17

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*135^q - 1) *

(m*135^q + 1)

odd n:

factor of 17

none - proven

27 (3250)

32 (2091)

1 (1171)

29 (697)

18 (569)

25 (317)

7 (26)

26 (13)

17 (11)

23 (6)

k = 16 proven composite by partial algebraic factors.

136

22195

3, 7, 43, 137

All k where k = m^2

and m = = 37 or 100 mod 137:

for even n let k = m^2

and let n = 2*q; factors to:

(m*136^q - 1) *

(m*136^q + 1)

odd n:

factor of 137

testing not started

testing not started

k = 1369 and 10000 proven composite by partial algebraic factors.

137

17

2, 3

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*137^q - 1) *

(m*137^q + 1)

odd n:

factor of 2

11, 13, 15 (all at n=2K)

16 (231)

3 (27)

5 (12)

1 (11)

10 (5)

14 (4)

12 (2)

8 (2)

2 (2)

7 (1)

k = 9 proven composite by partial algebraic factors.

138

1806

5, 13, 139

408, 688, 831, 1074, 1743 (all at n=300K)

421 (272919)

773 (249730)

372 (103160)

1368 (66926)

1087 (55582)

1258 (54256)

557 (52295)

359 (47249)

291 (35886)

9 (35685)

139

6

5, 7

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*139^q - 1) *

(m*139^q + 1)

odd n:

factor of 5

none - proven

1 (163)

3 (114)

5 (1)

2 (1)

k = 4 proven composite by partial algebraic factors.

140

46

3, 47

none - proven

38 (448)

11 (108)

1 (79)

5 (30)

29 (18)

32 (16)

14 (16)

33 (12)

40 (9)

41 (8)

141

285

2, 71

none - proven

201 (5279)

93 (1860)

197 (1052)

133 (818)

16 (573)

203 (250)

283 (244)

73 (237)

147 (209)

144 (171)

142

12

11, 13

none - proven

1 (1231)

3 (26)

11 (14)

8 (7)

6 (3)

4 (3)

10 (2)

9 (1)

7 (1)

5 (1)

143

5

2, 3

none - proven

3 (16)

1 (3)

2 (2)

4 (1)

144

59

5, 29

All k = m^2 for all n;

factors to:

(m*12^n - 1) *

(m*12^n + 1)

none - proven

39 (964)

30 (519)

23 (134)

46 (97)

58 (35)

2 (24)

57 (20)

15 (10)

54 (8)

34 (8)

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

145

1169

2, 73

(Condition 1):

All k where k = m^2

and m = = 27 or 46 mod 73:

for even n let k = m^2

and let n = 2*q; factors to:

(m*145^q - 1) *

(m*145^q + 1)

odd n:

factor of 73

(Condition 2):

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*145^q - 1) *

(m*145^q + 1)

odd n:

factor of 2

72, 113, 181, 303, 450, 523, 673, 769, 865, 1094, 1160 (all at n=2K)

8 (6368)

863 (1480)

838 (1460)

257 (1269)

1025 (1223)

347 (737)

817 (730)

641 (723)

685 (589)

759 (575)

k = 729 proven composite by condition 1.

k = 49, 81, 529, and 625 proven composite by condition 2.

146

8

3, 7

none - proven

5 (30)

2 (16)

1 (7)

4 (5)

3 (3)

6 (2)

7 (1)

147

73

2, 37

All k where k = m^2

and m = = 6 or 31 mod 37:

for even n let k = m^2

and let n = 2*q; factors to:

(m*147^q - 1) *

(m*147^q + 1)

odd n:

factor of 37

49, 51, 55, 58, 59, 63 (all at n=2K)

11 (2042)

33 (619)

64 (169)

19 (140)

38 (131)

71 (114)

12 (112)

48 (96)

22 (48)

15 (46)

k = 36 proven composite by partial algebraic factors.

148

1936

5, 13, 149

All k where k = m^2

and m = = 44 or 105 mod 149:

for even n let k = m^2

and let n = 2*q; factors to:

(m*148^q - 1) *

(m*148^q + 1)

odd n:

factor of 149

215, 256, 304, 346, 367, 448, 577, 580, 595, 636, 691, 694, 746, 801, 831, 898, 934, 967, 1015, 1048, 1052, 1134, 1204, 1234, 1249, 1256, 1258, 1307, 1341, 1351, 1426, 1489, 1516, 1594, 1600, 1604, 1621, 1743, 1750, 1852, 1901 (all at n=2K)

1554 (1991)

1312 (1967)

1381 (1942)

597 (1895)

417 (1891)

1357 (1890)

541 (1762)

281 (1738)

1228 (1657)

1841 (1586)

No k's proven composite by algebraic factors.

149

4

3, 5

none - proven

1 (7)

2 (4)

3 (1)

150

49074

7, 31, 103, 151

206, 841, 1509, 1962, 3229, 4682, 5245, 5890, 6039, 6353, 6494, 7851, 9061, 9260, 11324, 11477, 11516, 12839, 14373, 16309, 16404, 16424, 16977, 17603, 18859, 19027, 19191, 19226, 20468, 20988, 22238, 22349, 22977, 23396, 23706, 23944, 24614, 24852, 25488, 25704, 25829, 26685, 27032, 28389, 28822, 30050, 30993, 31738, 31812, 33521, 34429, 34707, 35066, 35344, 36709, 36994, 37137, 39108, 39141, 39712, 39736, 40020, 42012, 42128, 43060, 43789, 44346, 44645, 44832, 46257, 46616, 47717, 48138 (k = 30993 and 31738 at n=2K, other k at n=100K)

17554 (99646)

32797 (97430)

32399 (96963)

37966 (96107)

10505 (93910)

42643 (93875)

5674 (92155)

6492 (90168)

32135 (90000)

31409 (89441)

151

37

2, 19

9, 25 (both at n=2K)

3 (716)

34 (45)

29 (25)

22 (20)

4 (15)

27 (14)

1 (13)

16 (9)

13 (9)

23 (8)

152

16

3, 17

none - proven (with probable primes that have not been certified: k = 1)

14 (343720)

1 (270217)

2 (796)

13 (23)

11 (14)

5 (12)

10 (5)

3 (3)

15 (2)

8 (2)

153

34

7, 11

(Condition 1):

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*153^q - 1) *

(m*153^q + 1)

odd n:

factor of 2

(Condition 2):

All k where k = 17*m^2

and m = = 1 or 7 mod 8:

even n:

factor of 2

for odd n let k = 17*m^2 and let n=2*q-1; factors to:

[m*3^(2q-1)*17^q - 1] * [m*3^(2q-1)*17^q + 1]

none - proven

12 (21659)

21 (70)

27 (44)

22 (23)

32 (8)

15 (5)

20 (4)

4 (3)

1 (3)

30 (2)

k = 9 and 25 proven composite by condition 1.

k = 17 proven composite by condition 2.

154

61

5, 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*154^q - 1) *

(m*154^q + 1)

odd n:

factor of 5

none - proven

6 (1989)

39 (326)

19 (324)

24 (106)

14 (78)

29 (62)

54 (30)

36 (7)

31 (7)

21 (7)

k = 4, 9, and 49 proven composite by partial algebraic factors.

155

5

2, 3

none - proven

1 (3)

3 (2)

2 (2)

4 (1)

156

unknown (>10^9, <=2113322677)

unknown

(Condition 1):

All k where k = m^2

and m = = 28 or 129 mod 157:

for even n let k = m^2

and let n = 2*q; factors to:

(m*156^q - 1) *

(m*156^q + 1)

odd n:

factor of 157

(Condition 2):

All k where k = 39*m^2

and m = = 56 or 101 mod 157:

even n:

factor of 157

for odd n let k = 39*m^2

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

[m*2^(2q-1)*39^q - 1] *

[m*2^(2q-1)*39^q + 1]

testing not started

testing not started

k = 28^2, 129^2, 185^2, 286^2 (etc. pattern repeating every 157m) proven composite by condition 1.

k = 39*56^2, 39*101^2, 39*213^2, 39*258^2 (etc. pattern repeating every 157m) proven composite by condition 2.

157

17

2, 5, 29

none - proven

8 (56)

15 (49)

4 (45)

7 (32)

1 (17)

13 (10)

14 (7)

16 (5)

5 (4)

12 (2)

158

52

3, 53

29, 44 (both at n=500K)

47 (273942)

34 (5223)

46 (147)

41 (94)

38 (74)

39 (49)

7 (39)

9 (35)

20 (34)

8 (20)

159

9

2, 5

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*159^q - 1) *

(m*159^q + 1)

odd n:

factor of 5

none - proven

3 (2160)

8 (22)

1 (13)

7 (6)

6 (1)

5 (1)

2 (1)

k = 4 proven composite by partial algebraic factors.

160

22

7, 23

none - proven

20 (7570)

12 (11)

6 (8)

1 (7)

5 (3)

4 (3)

13 (2)

10 (2)

2 (2)

21 (1)

161

65

2, 3

none - proven

52 (549)

50 (328)

32 (316)

2 (228)

55 (153)

49 (103)

40 (67)

53 (46)

59 (36)

20 (26)

162

3259

5, 163, 181

274, 302, 456, 1205, 1358, 1588, 1828, 2118, 2178, 2297, 2423, 2703, 2841, 2997, 3144, 3249 (k = 2118 and 2841 at n=300K, other k at n=2K)

2018 (194314)

2954 (95124)

1308 (82803)

1607 (28018)

58 (13758)

2809 (12303)

423 (8898)

3098 (8723)

653 (8335)

1781 (8327)

163

81

2, 41

11, 37, 39, 57, 64 (all at n=2K)

4 (2285)

45 (1863)

75 (1000)

41 (955)

42 (775)

46 (249)

2 (84)

29 (37)

63 (36)

72 (24)

164

4

3, 5

none - proven

1 (3)

2 (2)

3 (1)

165

79

7, 13, 43

65 (15K)

53 (1174)

45 (184)

49 (171)

6 (86)

44 (71)

60 (67)

50 (41)

78 (29)

16 (17)

41 (13)

166

4174

3, 7, 13, 167

79, 187, 196, 222, 322, 337, 387, 424, 472, 556, 565, 571, 610, 615, 640, 759, 888, 946, 982, 1033, 1057, 1087, 1249, 1321, 1550, 1609, 1759, 1846, 1849, 1942, 1963, 2003, 2047, 2071, 2096, 2152, 2170, 2302, 2313, 2362, 2501, 2526, 2554, 2566, 2588, 2614, 2673, 2809, 3166, 3234, 3349, 3418, 3467, 3481, 3493, 3501, 3502, 3508, 3526, 3541, 3642, 3736, 3899, 3962, 3991, 4006, 4134 (all at n=2K)

3106 (1861)

1969 (1823)

1789 (1796)

1602 (1770)

4042 (1732)

823 (1698)

919 (1651)

3424 (1597)

2802 (1583)

2929 (1528)

167

5

2, 3

none - proven

4 (1865)

2 (8)

3 (6)

1 (3)

168

4744

5, 13, 17, 73

(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*168^q - 1) *

(m*168^q + 1)

odd n:

factor of 13

(Condition 2):

All k where k = 42*m^2

and m = = 3 or 10 mod 13:

even n:

factor of 13

for odd n let k = 42*m^2

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

[m*2^(2q-1)*42^q - 1] *

[m*2^(2q-1)*42^q + 1]

53, 495, 584, 586, 948, 1364, 1416, 1429, 1512, 1626, 1741, 1743, 1754, 1938, 2172, 2237, 2263, 2599, 2627, 2848, 2852, 3067, 3106, 3119, 3238, 3314, 3407, 3574, 3678, 3769, 3795, 3797, 3844, 4016, 4328, 4382, 4549, 4614, 4642, 4668, 4707, 4723 (k = 2172 at n=2K, other k at n=100K)

1689 (68676)

3309 (63795)

4471 (54466)

4185 (53498)

2846 (50670)

1717 (38259)

1829 (34296)

2885 (34186)

2942 (33546)

2523 (31457)

k = 25, 64, 324, 441, 961, 1156, 1936, 2209, 3249, and 3600 proven composite by condition 1.

k = 378 and 4200 proven composite by condition 2.

169

16

5, 17

All k = m^2 for all n;

factors to:

(m*13^n - 1) *

(m*13^n + 1)

none - proven

14 (2)

13 (2)

3 (2)

15 (1)

12 (1)

11 (1)

10 (1)

8 (1)

7 (1)

6 (1)

k = 1, 4, and 9 proven composite by full algebraic factors.

170

20

3, 19

none - proven

2 (166428)

8 (15422)

18 (360)

11 (108)

5 (38)

1 (17)

13 (13)

9 (7)

7 (3)

4 (3)

171

85

2, 43

15, 51, 75 (all at n=2K)

5 (2925)

1 (181)

11 (138)

68 (83)

42 (72)

7 (68)

3 (60)

73 (51)

61 (45)

23 (32)

172

235

3, 7, 13

22, 127, 133, 184, 219 (k = 219 at n=300K, other k at n=2K)

30 (1160)

196 (749)

164 (603)

139 (573)

200 (468)

230 (231)

148 (103)

103 (95)

100 (89)

217 (80)

173

13

2, 3

11 (6K)

5 (54)

7 (15)

2 (4)

10 (3)

1 (3)

12 (2)

8 (2)

6 (2)

3 (2)

9 (1)

174

6

5, 7

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*174^q - 1) *

(m*174^q + 1)

odd n:

factor of 5

none - proven

1 (3251)

5 (2)

3 (1)

2 (1)

k = 4 proven composite by partial algebraic factors.

175

21

2, 11

none - proven

11 (3048)

10 (136)

3 (90)

16 (17)

5 (13)

18 (10)

15 (8)

14 (7)

1 (5)

19 (2)

176

58

3, 59

none - proven

34 (79)

26 (20)

22 (19)

53 (16)

50 (12)

32 (12)

29 (12)

25 (9)

4 (9)

43 (7)

177

209

2, 5, 13

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*177^q - 1) *

(m*177^q + 1)

odd n:

factor of 2

25, 161, 193, 197 (all at n=2K)

64 (340147)

36 (2957)