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Riesel conjectures

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Definition

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For the original Riesel problem, it is finding and proving the smallest k such that k×bn-1 is not prime for all integers n ≥ 1 and GCD(k-1, b-1)=1.

Extended definiton

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Finding and proving the smallest k such that (k×bn-1)/GCD(k-1, b-1) is not prime for all integers n ≥ 1.

Notes

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All n must be >= 1.

k-values that make a full covering set with all or partial algebraic factors are excluded from the conjectures.

k-values that are a multiple of base (b) and where (k-1)/gcd(k-1,b-1) is not prime are included in the conjectures but excluded from testing.

Such k-values will have the same prime as k / b.

Table

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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 (primality certificate for k=106) 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) (the k=139 prime is proven prime by N-1, and primality certificate for the large prime factor of N-1) (primality certificate for k=367, primality certificate for k=313, primality certificate for k=159, primality certificate for k=429, primality certificate for k=391, primality certificate for k=299, [http://factordb.com/cert.php?id=1100000000854476434 primality certificate for k=79) 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 (primality certificate for k=121) 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 (primality certificate for k=298) 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 (primality certificate for k=29, primality certificate for k=13) 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 (primality certificate for k=121) 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 (for the k=1 prime, factor N-1 is equivalent to factor 35^312-1) 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 (for the k=5 prime, factor N-1 is equivalent to factor 37^900-1) 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 (for the k=1 prime, factor N-1 is equivalent to factor 39^348-1) 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 (primality certificate for k=53) 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 (primality certificate for k=79) 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 (primality certificate for k=1) 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 (the k=87 prime is proven prime by N-1, and primality certificate for the large prime factor of N-1) 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 (primality certificate for k=25) 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 (primality certificate for k=79, primality certificate for k=101) 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 (primality certificate for k=35) 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 (primality certificate for k=53) 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 (primality certificate for k=19) 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 (primality certificate for k=27, primality certificate for k=1, primality certificate for k=37) 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 (for the k=1 prime, factor N-1 is equivalent to factor 92^438-1) (primality certificate for k=29) 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 (primality certificate for k=133) 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 (for the k=5 prime, factor N-1 is equivalent to factor 101^350-1) 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 (primality certificate for k=19, primality certificate for k=22, primality certificate for k=23) 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 (primality certificate for k=3) 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 (primality certificate for k=79) 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 (for the k=1 prime, factor N-1 is equivalent to factor 135^1170-1) (the k=25 prime is proven prime by N-1, and primality certificate for the large prime factor of N-1) (primality certificate for k=27, primality certificate for k=29) 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 (for the k=1 prime, factor N-1 is equivalent to factor 139^162-1) 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 (primality certificate for k=201, primality certificate for k=93, primality certificate for k=197, primality certificate for k=133, primality certificate for k=16, primality certificate for k=203, primality certificate for k=283, primality certificate for k=73, primality certificate for k=147) 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 (primality certificate for k=1) 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 (primality certificate for k=19) 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 (primality certificate for k=3) 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 (primality certificate for k=55) 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 (primality certificate for k=1) 1 (3251)

5 (2)

3 (1)

2 (1)

k = 4 proven composite by partial algebraic factors.
175 21 2, 11 none - proven (the k=10 prime is proven prime by N+1, and for the large prime factor of N+1, factor N-1 is equivalent to factor 175^136-1) (primality certificate for k=11) 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)

44 (1711)

163 (963)

97 (609)

33 (431)

179 (383)

200 (288)

58 (219)

172 (200)

k = 49 and 81 proven composite by partial algebraic factors.
178 22 3, 5, 7, 13, 97 4 (13K) 19 (13655)

11 (177)

6 (118)

21 (89)

14 (44)

3 (14)

17 (12)

13 (8)

7 (4)

16 (3)

179 4 3, 5 none - proven 1 (19)

3 (16)

2 (2)

180 7674582 7, 31, 181, 1051 (Condition 1):

All k where k = m^2

and m = = 19 or 162 mod 181:

for even n let k = m^2

and let n = 2*q; factors to:

(m*180^q - 1) *

(m*180^q + 1)

odd n:

factor of 181

(Condition 2):

All k where k = 5*m^2

and m = = 67 or 114 mod 181:

even n:

factor of 181

for odd n let k = 5*m^2

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

[m*6^(2q-1)*5^q - 1] *

[m*6^(2q-1)*5^q + 1]

testing not started testing not started k = 19^2, 162^2, 200^2, 343^2 (etc. pattern repeating every 181m) proven composite by condition 1.

k = 5*67^2, 5*114^2, 5*248^2, 5*295^2 (etc. pattern repeating every 181m) proven composite by condition 2.

181 25 2, 13 5, 21 (k = 5 at n=21K, k = 21 at n=12K) 14 (29)

1 (17)

12 (8)

24 (5)

10 (5)

9 (5)

15 (3)

20 (2)

13 (2)

6 (2)

182 62 3, 61 none - proven (for the k=1 prime, factor N-1 is equivalent to factor 182^166-1) 43 (502611)

26 (990)

29 (632)

54 (329)

7 (209)

1 (167)

44 (152)

58 (127)

47 (122)

59 (96)

183 45 2, 23 none - proven (for the k=1 prime, factor N-1 is equivalent to factor 183^222-1) (the k=37 prime is proven prime by N-1, and primality certificate for the large prime factor of N-1) (primality certificate for k=13, primality certificate for k=23, primality certificate for k=17) 13 (581)

23 (534)

1 (223)

17 (175)

37 (155)

15 (42)

27 (40)

26 (37)

21 (27)

42 (11)

184 36 5, 37 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*184^q - 1) *

(m*184^q + 1)

odd n:

factor of 5

none - proven (with probable primes that have not been certified: k = 1) 1 (16703)

28 (85)

7 (32)

16 (21)

11 (15)

19 (10)

24 (8)

14 (8)

22 (7)

34 (6)

k = 4 and 9 proven composite by partial algebraic factors.
185 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*185^q - 1) *

(m*185^q + 1)

odd n:

factor of 2

1 (66.3K) 10 (6783)

12 (8)

8 (8)

14 (4)

11 (4)

5 (4)

16 (3)

15 (2)

2 (2)

13 (1)

k = 9 proven composite by partial algebraic factors.
186 67 11, 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*186^q - 1) *

(m*186^q + 1)

odd n:

factor of 17

36 (13K) 12 (112717)

32 (388)

43 (44)

51 (32)

44 (14)

35 (13)

52 (11)

58 (9)

42 (7)

1 (7)

k = 16 proven composite by partial algebraic factors.
187 51 2, 5, 13 13, 27, 33, 39 (all at n=2K) 17 (1125)

7 (510)

43 (136)

11 (110)

31 (74)

48 (71)

1 (37)

10 (16)

18 (12)

23 (10)

188 8 3, 7 none - proven 6 (950)

5 (40)

7 (7)

1 (3)

2 (2)

4 (1)

3 (1)

189 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*189^q - 1) *

(m*189^q + 1)

odd n:

factor of 5

none - proven 6 (3)

2 (3)

1 (3)

5 (2)

8 (1)

7 (1)

3 (1)

k = 4 proven composite by partial algebraic factors.
190 626861 13, 89, 191, 1753 testing not started testing not started
191 5 2, 3 none - proven 2 (970)

1 (17)

4 (5)

3 (2)

192 13897 5, 73, 193 All k where k = m^2

and m = = 81 or 112 mod 193:

for even n let k = m^2

and let n = 2*q; factors to:

(m*192^q - 1) *

(m*192^q + 1)

odd n:

factor of 193

253, 311, 593, 894, 898, 1268, 1422, 1704, 2118, 2264, 2315, 2324, 2396, 2441, 2909, 3092, 3282, 3303, 3323, 3719, 3859, 4038, 4062, 4078, 4104, 4164, 4247, 4304, 4372, 4426, 4618, 4679, 5132, 5173, 5523, 5547, 5584, 5731, 5758, 5761, 5789, 5967, 5984, 6083, 6175, 6177, 6205, 6261, 6263, 6297, 6353, 6354, 6484, 6547, 6558, 6746, 6789, 6889, 6939, 7096, 7407, 7528, 7549, 7591, 7756, 7889, 7913, 7931, 7984, 8187, 8214, 8248, 8347, 8361, 8382, 8493, 8537, 8988, 9091, 9111, 9208, 9402, 9689, 9883, 10037, 10063, 10162, 10349, 10396, 10423, 10488, 10657, 10817, 10988, 11002, 11213, 11488, 11933, 12132, 12157, 12234, 12317, 12424, 12716, 12782, 12797, 12906, 12983, 12984, 13358, 13484, 13605, 13623, 13738, 13798 (k = 5731 and 8214 at n=2K, other k at n=100K) 10909 (89859)

2486 (88582)

49 (88335)

2258 (86531)

7511 (85174)

12732 (85108)

12807 (84820)

9344 (83216)

1023 (78795)

2423 (77515)

k = 6561 and 12544 proven composite by partial algebraic factors.
193 484 3, 5, 7, 13, 97 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*193^q - 1) *

(m*193^q + 1)

odd n:

factor of 97

30, 58, 95, 106, 116, 134, 169, 184, 207, 226, 272, 302, 348, 379, 449, 463 (all at n=2K) 466 (1986)

431 (1794)

297 (1700)

387 (1638)

93 (1473)

136 (1018)

121 (849)

408 (725)

256 (417)

135 (413)

No k's proven composite by algebraic factors.
194 4 3, 5 none - proven 2 (42)

3 (3)

1 (3)

195 13 2, 7 none - proven 6 (38)

1 (11)

11 (4)

4 (3)

7 (2)

3 (2)

12 (1)

10 (1)

9 (1)

8 (1)

196 1267 3, 61, 211 All k = m^2 for all n;

factors to:

(m*14^n - 1) *

(m*14^n + 1)

198, 202, 223, 423, 562, 617, 647, 735, 808, 976, 1183 (all at n=2K) 5 (9849)

947 (1797)

807 (1630)

973 (1574)

342 (1548)

1111 (1455)

865 (649)

877 (639)

1087 (541)

962 (485)

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.
197 10 3, 11 none - proven (primality certificate for k=7) 7 (249)

1 (31)

5 (10)

8 (4)

3 (4)

2 (2)

9 (1)

6 (1)

4 (1)

198 3662 7, 13, 433 81, 172, 424, 464, 484, 529, 991, 1037, 1054, 1262, 1283, 1792, 1856, 1920, 2253, 2272, 2304, 2445, 2577, 2787, 2811, 2934, 3103, 3207, 3305, 3329, 3342, 3602, 3649 (all at n=100K) 2661 (95399)

1284 (73379)

807 (50662)

2791 (48837)

2187 (43879)

2388 (43718)

848 (40132)

947 (36807)

3420 (35891)

1922 (31592)

199 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*199^q - 1) *

(m*199^q + 1)

odd n:

factor of 5

none - proven (for the k=1 prime, factor N-1 is equivalent to factor 199^576-1) 1 (577)

7 (104)

3 (24)

8 (5)

5 (3)

6 (1)

2 (1)

k = 4 proven composite by partial algebraic factors.
200 68 3, 67 none - proven (with probable primes that have not been certified: k = 1) 38 (131900)

58 (102363)

53 (45666)

51 (44252)

23 (31566)

19 (29809)

1 (17807)

13 (12053)

37 (597)

62 (126)

256 100 3, 7, 13 All k = m^2 for all n;

factors to:

(m*16^n - 1) *

(m*16^n + 1)

none - proven 74 (319)

47 (228)

42 (224)

92 (143)

68 (87)

61 (54)

35 (28)

65 (24)

70 (18)

75 (17)

k = 1, 4, 9, 16, 25, 36, 49, 64, and 81 proven composite by full algebraic factors.
512 14 3, 5, 13 All k = m^3 for all n;

factors to:

(m*8^n - 1) *

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

none - proven 4 (2215)

13 (2119)

9 (7)

11 (6)

6 (6)

5 (2)

3 (2)

2 (2)

12 (1)

10 (1)

k = 1 and 8 proven composite by full algebraic factors.
1024 81 5, 41 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 (k = 29 at n=1M, other k at n=3K) 74 (666084)

39 (4070)

43 (2290)

13 (1167)

78 (424)

65 (93)

69 (54)

3 (47)

71 (41)

44 (36)

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