Riesel conjectures

## 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

[edit | edit source]

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.