Siepinski conjectures

## Definition

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For the original Sierpinski 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

[edit | edit source]

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 Sierpinski 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  | 78557  | 3, 5, 7, 13, 19, 37, 73  |  | 21181, 22699, 24737, 55459, 65536, 67607 (k = 65536 at n=8.589G, other k at n=37M)  | 10223 (31172165)

19249 (13018586)

27653 (9167433)

28433 (7830457)

33661 (7031232)

5359 (5054502)

4847 (3321063)

54767 (1337287)

69109 (1157446)

65567 (1013803)

3  | 11047  | 2, 5, 7, 13, 73  |  | 1187, 1801, 3007, 3047, 3307, 5321, 5743, 5893, 6427, 6569, 6575, 7927, 8161, 8227, 8467, 8609, 8863, 8987, 9263, 9449 (all at n=16.3K)  | 621 (20820)

3061 (15772)

10243 (9731)

2747 (7097)

10207 (6089)

823 (6087)

10741 (6028)

821 (5512)

5147 (5153)

9721 (5040)

4  | 419  | 3, 5, 7, 13  |  | none - proven  | 186 (10458)

94 (291)

176 (228)

129 (207)

89 (167)

86 (108)

174 (103)

369 (71)

101 (66)

293 (58)

5  | 7  | 2, 3  |  | none - proven  | 4 (2)

3 (2)

6 (1)

5 (1)

2 (1)

1 (1)

6  | 174308  | 7, 13, 31, 37, 97  |  | 1296, 1814, 9589, 12179, 13215, 14505, 22139, 23864, 29014, 43429, 49874, 50252, 57189, 62614, 67894, 73814, 76441, 80389, 87284, 87289, 87800, 97131, 100899, 112783, 117454, 122704, 124874, 127688, 132614, 135199, 139959, 145984, 151719, 152209, 166753, 168610 (k = 1296 at n=268.4M, k = 1814 at n=200K, other k = 4 mod 5 at n=33.5K, other k at n=4M)  | 124125 (2018254)

139413 (1279992)

33706 (910462)

125098 (896696)

31340 (833096)

59506 (780877)

10107 (559967)

113966 (511831)

172257 (349166)

121736 (298935)

7  | 209  | 2, 3, 5, 13, 43  |  | none - proven (primality certificate for k=141)  | 141 (1044)

121 (252)

101 (216)

21 (124)

181 (80)

173 (48)

87 (47)

145 (46)

77 (44)

187 (35)

8  | 47  | 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  | 31 (20)

46 (4)

40 (4)

37 (4)

28 (4)

16 (4)

13 (4)

45 (3)

38 (3)

36 (3)

| k = 1, 8, and 27 proven composite by full algebraic factors.
9  | 31  | 2, 5  |  | none - proven  | 26 (6)

21 (4)

24 (3)

17 (3)

28 (2)

23 (2)

16 (2)

11 (2)

10 (2)

7 (2)

10  | 989  | 3, 7, 11, 13  |  | 100, 269 (k = 100 at n=2.147G, k = 269 at n=100K)  | 804 (5470)

342 (338)

485 (230)

912 (215)

815 (190)

378 (188)

494 (135)

640 (120)

737 (117)

603 (107)

11  | 5  | 2, 3  |  | none - proven  | 4 (2)

1 (2)

3 (1)

2 (1)

12  | 521  | 5, 13, 29  |  | 12 (33.55M)  | 404 (714558)

378 (2388)

261 (644)

407 (367)

354 (291)

37 (199)

30 (144)

88 (113)

17 (78)

274 (74)

13  | 15  | 2, 7  |  | none - proven (for the k=11 prime, factor N-1 is equivalent to factor 13^564-1)  | 11 (564)

8 (4)

13 (3)

3 (2)

2 (2)

14 (1)

12 (1)

10 (1)

9 (1)

7 (1)

14  | 4  | 3, 5  |  | none - proven  | 1 (2)

3 (1)

2 (1)

15  | 673029  | 2, 17, 113, 1489  |  | 225, 341, 343, 641, 965, 1205, 1827, 2263, 2323, 2403, 2445, 2461, 2471, 2531, 2813, 3347, 3625, 3797, 3935, 3959, 4045, 4169, 4355, 4665, 4733, 5169, 5793, 5891, 5983, 6061, 6331, 6553, 6661, 6775, 6849, 7087, 7693, 7711, 7773, 7975, 7979, 8017, 8161, 8181, 8271, 8603, 8881, 9215, 9643, 9767, 9783, 9857 (for k <= 10K) (k = 225 at n=524K, other k at n=1.5K)  | 6598 (11715)

6476 (1522)

5529 (1446)

6313 (1276)

7763 (1179)

4787 (1129)

219 (1129)

5975 (1099)

7957 (1082)

5653 (1064)

16  | 38  | 3, 7, 13  | All k=4*q^4 for all n:

let k=4*q^4

and let m=q*2^n; factors to:

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

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

| none - proven (primality certificate for k=23)  | 23 (1074)

33 (7)

35 (4)

18 (4)

10 (3)

5 (3)

32 (2)

31 (2)

30 (2)

24 (2)

| k = 4 proven composite by full algebraic factors.
17  | 31  | 2, 3  |  | none - proven  | 10 (1356)

7 (190)

2 (47)

29 (41)

20 (13)

23 (9)

4 (6)

16 (4)

1 (4)

30 (3)

18  | 398  | 5, 13, 19  |  | 18 (33.55M)  | 122 (292318)

381 (24108)

291 (2415)

37 (457)

362 (258)

123 (236)

183 (171)

363 (163)

209 (79)

318 (78)

19  | 9  | 2, 5  |  | none - proven  | 5 (78)

6 (14)

4 (3)

1 (2)

8 (1)

7 (1)

3 (1)

2 (1)

20  | 8  | 3, 7  |  | none - proven  | 6 (15)

7 (2)

4 (2)

1 (2)

5 (1)

3 (1)

2 (1)

21  | 23  | 2, 11  |  | none - proven  | 12 (10)

21 (3)

19 (2)

11 (2)

8 (2)

3 (2)

22 (1)

20 (1)

18 (1)

17 (1)

22  | 2253  | 5, 23, 97  |  | 22, 1754, 1772, 1862, 2186, 2232 (k = 22 at n=16.77M, other k at n=16.8K)  | 1611 (738988)

1908 (355313)

942 (18359)

740 (18137)

1496 (17480)

461 (16620)

953 (5596)

1793 (4121)

1161 (3720)

346 (3180)

23  | 5  | 2, 3  |  | none - proven  | 4 (342)

1 (4)

3 (3)

2 (1)

24  | 30651  | 5, 7, 13, 73, 79  |  | 656, 1099, 1816, 1851, 1864, 2164, 2351, 2529, 2586, 3404, 3526, 3609, 4346, 4606, 4894, 5129, 5316, 5324, 5386, 5889, 5974, 7276, 7746, 7844, 8054, 8091, 8161, 9279, 9304, 9701, 9721, 10026, 10156, 10326, 10531, 11346, 12626, 12969, 12991, 13716, 14006, 14604, 15921, 17334, 17819, 17876, 18006, 18204, 18911, 19031, 19094, 20219, 20676, 20731, 21459, 21849, 22289, 22356, 22479, 23844, 23874, 24784, 25964, 25966, 26279, 27344, 29091, 29349, 29464, 29566, 29601 (k = 22 mod 23 at n=11.3K, other k at n=400K)  | 13984 (397259)

3846 (383526)

23981 (360062)

8369 (359371)

3706 (353908)

12799 (353083)

29009 (338099)

28099 (332519)

21526 (329368)

26804 (266195)

25  | 79  | 2, 13  |  | 71 (10K)  | 61 (3104)

40 (518)

59 (48)

77 (27)

68 (15)

47 (9)

12 (9)

51 (7)

66 (6)

57 (5)

26  | 221  | 3, 7, 19, 37  |  | 65, 155 (both at n=1M)  | 32 (318071)

217 (11454)

95 (1683)

178 (1154)

138 (827)

157 (308)

175 (276)

211 (98)

149 (87)

197 (71)

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 (10)

7 (3)

12 (2)

5 (2)

2 (2)

11 (1)

10 (1)

6 (1)

4 (1)

3 (1)

| k = 1 and 8 proven composite by full algebraic factors.
28  | 4554  | 5, 29, 157  |  | 871, 3104, 4552 (k = 3104 at n=25.5K, k = 871 and 4552 at n=1M)  | 3394 (427262)

4233 (331135)

2377 (104621)

146 (47316)

1291 (22811)

2203 (13911)

1565 (8607)

1797 (5681)

1043 (5459)

2467 (4956)

29  | 4  | 3, 5  |  | none - proven  | 3 (2)

1 (2)

2 (1)

30  | 867  | 7, 13, 19, 31  |  | 278, 588 (both at n=1M)  | 699 (11837)

242 (5064)

659 (4936)

311 (1760)

559 (1654)

557 (1463)

740 (1135)

12 (1023)

83 (644)

293 (361)

31  | 239  | 2, 3, 7, 19  |  | 1, 51, 73, 77, 107, 117, 149, 181, 209 (k = 1 at n=524K, k = 51 at n=37K, other k at n=6K)  | 43 (21053)

189 (5570)

191 (1553)

5 (1026)

113 (178)

121 (118)

145 (78)

37 (64)

33 (62)

205 (60)

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)

| 4 (1.717G)  | 9 (13)

7 (4)

5 (3)

2 (3)

8 (1)

6 (1)

3 (1)

| k = 1 proven composite by full algebraic factors.
33  | 511  | 2, 17  |  | 67, 203 (both at n=12K)  | 36 (23615)

407 (10961)

154 (6846)

319 (5043)

288 (4583)

418 (780)

11 (593)

305 (561)

251 (495)

63 (347)

34  | 6  | 5, 7  |  | none - proven  | 5 (12)

1 (4)

4 (1)

3 (1)

2 (1)

35  | 5  | 2, 3  |  | none - proven  | 4 (42)

1 (2)

3 (1)

2 (1)

36  | 1886  | 13, 31, 37, 43  |  | 1296, 1814 (k = 1296 at n=134.2M, k = 1814 at n=100K)  | 960 (1571)

716 (1554)

526 (698)

1000 (542)

223 (480)

1096 (407)

1570 (352)

667 (302)

1115 (280)

1669 (240)

37  | 39  | 2, 19  |  | 37 (524K)  | 19 (5310)

18 (461)

17 (12)

36 (9)

35 (6)

33 (6)

3 (6)

31 (5)

32 (4)

11 (4)

38  | 14  | 3, 13  |  | 1 (16.77M)  | 2 (2729)

9 (21)

4 (10)

8 (7)

10 (4)

7 (4)

3 (3)

13 (2)

12 (1)

11 (1)

39  | 9  | 2, 5  |  | none - proven  | 6 (2)

5 (2)

1 (2)

8 (1)

7 (1)

4 (1)

3 (1)

2 (1)

40  | 47723  | 3, 7, 41, 223  |  | 1169, 1229, 1415, 1600, 2215, 2294, 2338, 2543, 2789, 2951, 2957, 3050, 3281, 3689, 3812, 3935, 4224, 4388, 4468, 4565, 4675, 4742, 4820, 5003, 5042, 5126, 5372, 5944, 6689, 7051, 7092, 7586, 7934, 8255, 8283, 8362, 8363, 8792, 8978, 9090, 9101, 9221, 9224, 9731, 9964, 10187, 10661, 10762, 11112, 11195, 11438, 11645, 11684, 12422, 12668, 12955, 13025, 13193, 13283, 13406, 13445, 13970, 15104, 15263, 15284, 15374, 15579, 15581, 15989, 16235, 16319, 16445, 16481, 16768, 16850, 17465, 17477, 17957, 18146, 18164, 18285, 18365, 18572, 18692, 18695, 18818, 19202, 19213, 19280, 19394, 19884, 20124, 20198, 20267, 20318, 20870, 20894, 20951, 20963, 21032, 21196, 21407, 21895, 22671, 22961, 23123, 23201, 23371, 23741, 23984, 24221, 24437, 24476, 24594, 25667, 26198, 26387, 26815, 26855, 27182, 27389, 27430, 28332, 28496, 28578, 28619, 29045, 29108, 29150, 29291, 29603, 29642, 30236, 30269, 30503, 30505, 30751, 31079, 31088, 31220, 31226, 31489, 31538, 31770, 31928, 32512, 32555, 32637, 32678, 32717, 33065, 33211, 33344, 33662, 33764, 33785, 33929, 34029, 34646, 34709, 34808, 35333, 35375, 35382, 35384, 35417, 35507, 35546, 35552, 35822, 35828, 35837, 35894, 35999, 36101, 36185, 36368, 36824, 37229, 37268, 37577, 37703, 38324, 38828, 38951, 39115, 39230, 39722, 40667, 41411, 41450, 41479, 41696, 41819, 42106, 43174, 43295, 43787, 43830, 43892, 43994, 44238, 44279, 44546, 44732, 44894, 46370, 46698, 46709, 46925, 47272, 47276, 47559, 47684 (all at n=5K)  | 14555 (4988)

39119 (4945)

21026 (4919)

20402 (4907)

39525 (4904)

8624 (4892)

15417 (4860)

25501 (4717)

27948 (4710)

5477 (4683)

41  | 8  | 3, 7  |  | none - proven  | 1 (16)

4 (6)

6 (3)

7 (2)

5 (1)

3 (1)

2 (1)

42  | 13372  | 5, 43, 353  |  | 42, 988, 1117, 1421, 3226, 4127, 5503, 6707, 8298, 8601, 9074, 11093, 11717, 11738, 11912, 12256, 13283 (k = 42 at n=16.77M, k = 13283 at n=10K, other k at n=800K)  | 8343 (560662)

12001 (312245)

12042 (277646)

4643 (143933)

4297 (142044)

4731 (141968)

3897 (136780)

10009 (132629)

2794 (126595)

8300 (116404)

43  | 21  | 2, 11  |  | none - proven (for the k=13 prime, factor N-1 is equivalent to factor 43^580-1) (primality certificate for k=9)  | 13 (580)

9 (498)

3 (171)

5 (38)

17 (34)

15 (23)

1 (8)

18 (3)

16 (3)

14 (2)

44  | 4  | 3, 5  |  | none - proven  | 1 (16)

3 (9)

2 (1)

45  | 47  | 2, 23  |  | none - proven  | 24 (18522)

15 (55)

42 (36)

3 (28)

35 (22)

8 (8)

30 (5)

38 (3)

23 (3)

20 (3)

46  | 881  | 3, 7, 103  |  | 563, 845 (both at n=35K)  | 283 (21198)

17 (4920)

140 (2105)

619 (2005)

278 (1788)

347 (1287)

729 (1006)

95 (446)

229 (443)

871 (405)

47  | 5  | 2, 3  |  | none - proven  | 2 (175)

1 (8)

4 (2)

3 (1)

48  | 1219  | 7, 13, 61, 181  |  | 36, 62, 153, 561, 622, 1114, 1168 (all at n=500K)  | 937 (309725)

701 (284564)

1077 (216501)

1086 (136352)

1121 (133656)

29 (133042)

841 (84732)

1099 (81106)

359 (35671)

1028 (22619)

49  | 31  | 2, 5  |  | none - proven  | 24 (165)

21 (62)

22 (39)

11 (26)

16 (10)

29 (9)

9 (3)

26 (2)

20 (2)

15 (2)

50  | 16  | 3, 17  |  | 1 (16.77M)  | 7 (516)

4 (10)

11 (9)

10 (4)

13 (2)

9 (2)

15 (1)

14 (1)

12 (1)

8 (1)

51  | 25  | 2, 13  |  | none - proven  | 5 (6)

24 (5)

21 (4)

13 (4)

10 (3)

3 (3)

17 (2)

16 (2)

14 (2)

9 (2)

52  | 28674  | 5, 53, 541  |  | 52, 113, 158, 266, 311, 317, 419, 584, 674, 743, 863, 902, 973, 1043, 1292, 1376, 1483, 1502, 1538, 1591, 1658, 1727, 1808, 1907, 2174, 2384, 2386, 2570, 2624, 2711, 2813, 2978, 3181, 3232, 3254, 3418, 3671, 3746, 4133, 4135, 4241, 4292, 4706, 4901, 4928, 4967, 5281, 5282, 5405, 5570, 5573, 5619, 5624, 5693, 5711, 5776, 5882, 5988, 6011, 6125, 6147, 6149, 6239, 6536, 6572, 6687, 6770, 6891, 7058, 7089, 7147, 7207, 7262, 7283, 7313, 7397, 7400, 7577, 7580, 7737, 7739, 7998, 8054, 8638, 8681, 8693, 8990, 9083, 9134, 9243, 9329, 9356, 9421, 9433, 9437, 9602, 9737, 9848, 9943, 9977, 10004, 10013, 10188, 10246, 10328, 10451, 10487, 10493, 10499, 10548, 10586, 10601, 10641, 10652, 10667, 10679, 10739, 10916, 10919, 10999, 11078, 11146, 11516, 11553, 11684, 11714, 11747, 11771, 11798, 11818, 12191, 12197, 12461, 12471, 12533, 12721, 12779, 12918, 13043, 13171, 13251, 13277, 13514, 13673, 13697, 13784, 13799, 13842, 13952, 14132, 14256, 14849, 14888, 15110, 15157, 15282, 15422, 15424, 15474, 15636, 15637, 15659, 15901, 16058, 16133, 16273, 16535, 16559, 16738, 16749, 16802, 16853, 16961, 17012, 17027, 17054, 17120, 17277, 17279, 17383, 17491, 17712, 17723, 17809, 17996, 18072, 18328, 18449, 18458, 18526, 18602, 18632, 18797, 18816, 18951, 19043, 19081, 19121, 19157, 19178, 19241, 19319, 19352, 19397, 19403, 19451, 19493, 19556, 19646, 19721, 19751, 19768, 19959, 19980, 19982, 20192, 20351, 20459, 20475, 20526, 20722, 20840, 20897, 20936, 20975, 21246, 21272, 21347, 21353, 21359, 21517, 21851, 21902, 22055, 22169, 22332, 22418, 22430, 22526, 22701, 22709, 22719, 22739, 22791, 23062, 23531, 23558, 23586, 23612, 23663, 23705, 23743, 23774, 23844, 23871, 23902, 23987, 24257, 24273, 24328, 24347, 24452, 24456, 24464, 24547, 24563, 24697, 24866, 24911, 25227, 25229, 25236, 25439, 25492, 25494, 25653, 25704, 25865, 25943, 26078, 26261, 26287, 26498, 26658, 26660, 26744, 26771, 26858, 26923, 26966, 27082, 27122, 27327, 27527, 27572, 27623, 27877, 28142, 28193, 28198, 28462, 28493, 28661 (all at n=5K)  | 14129 (4891)

19634 (4877)

8132 (4875)

42 (4822)

3827 (4716)

15656 (4640)

6044 (4635)

21167 (4604)

10861 (4597)

20987 (4571)

53  | 7  | 2, 3  |  | 4 (2.075M)  | 6 (143)

5 (9)

1 (8)

3 (4)

2 (1)

54  | 21  | 5, 11  |  | none - proven  | 19 (103)

16 (30)

13 (7)

12 (4)

4 (3)

20 (2)

18 (2)

11 (2)

6 (2)

1 (2)

55  | 13  | 2, 7  |  | 1 (524K)  | 10 (9)

9 (2)

8 (2)

5 (2)

4 (2)

12 (1)

11 (1)

7 (1)

6 (1)

3 (1)

56  | 20  | 3, 19  |  | none - proven  | 4 (78)

19 (70)

13 (6)

7 (6)

3 (5)

16 (2)

15 (2)

10 (2)

1 (2)

18 (1)

57  | 47  | 2, 5, 13  |  | none - proven  | 14 (14955)

39 (74)

27 (44)

46 (20)

30 (14)

31 (7)

38 (5)

25 (5)

16 (5)

6 (5)

58  | 488  | 3, 7, 163  |  | 58, 122, 176, 222, 431, 437, 461 (k = 58 at n=16.77M, k = 222 at n=125K, other k at n=14.9K)  | 178 (25524)

297 (11508)

266 (9040)

241 (1964)

296 (1892)

393 (1831)

106 (1795)

228 (1603)

20 (1340)

392 (1222)

59  | 4  | 3, 5  |  | none - proven  | 2 (3)

1 (2)

3 (1)

60  | 16957  | 13, 61, 277  |  | 60, 853, 1646, 2075, 4025, 4406, 4441, 5064, 6772, 7262, 7931, 10226, 11406, 12323, 13785, 14958, 15007, 15452, 15676, 16050 (k = 60 at n=16.77M, other k at n=500K)  | 14066 (324990)

16014 (227010)

5767 (201439)

12927 (191870)

11441 (180105)

8923 (109088)

13846 (90979)

2497 (88149)

10405 (77541)

6465 (37209)

61  | 63  | 2, 31  |  | none - proven (primality certificate for k=62, primality certificate for k=43, primality certificate for k=23)  | 62 (3698)

43 (2788)

23 (1659)

10 (165)

19 (70)

32 (18)

25 (16)

36 (12)

57 (11)

26 (11)

62  | 8  | 3, 7  |  | 1 (16.77M)  | 7 (308)

2 (43)

3 (12)

4 (2)

6 (1)

5 (1)

63  | 1589  | 2, 5, 397  |  | 1, 83, 101, 103, 113, 143, 185, 223, 237, 267, 307, 309, 335, 343, 367, 381, 391, 411, 425, 467, 471, 487, 509, 549, 587, 603, 637, 643, 645, 673, 677, 681, 687, 689, 701, 789, 807, 821, 825, 827, 881, 903, 937, 951, 963, 983, 989, 1021, 1043, 1047, 1063, 1067, 1103, 1263, 1267, 1283, 1321, 1341, 1401, 1461, 1463, 1467, 1481, 1523, 1553, 1563, 1581 (k = 1 at n=524K, other k at n=2K)  | 1108 (12351)

888 (2698)

9 (2162)

1174 (1989)

1201 (1904)

1367 (1861)

1189 (1846)

1027 (1693)

581 (1596)

1433 (1554)

64  | 14  | 5, 13  | All k = m^3 for all n;

factors to:

(m*4^n + 1) *

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

| none - proven (primality certificate for k=11)  | 11 (3222)

13 (2)

6 (2)

12 (1)

10 (1)

9 (1)

7 (1)

5 (1)

4 (1)

3 (1)

| k = 1 and 8 proven composite by full algebraic factors.
65  | 10  | 3, 11  |  | none - proven  | 6 (5)

7 (2)

4 (2)

3 (2)

1 (2)

9 (1)

8 (1)

5 (1)

2 (1)

66  | 21314443  | 7, 17, 37, 67, 73, 4357  |  | 269, 470, 537, 1198, 1408, 1449, 2076, 2257, 2464, 2605, 2614, 2624, 2815, 3284, 3899, 4153, 4155, 4175, 4356, 4689, 4769, 4820, 4883, 5024, 5200, 5334, 5361, 5442, 5765, 5805, 5857, 6031, 6289, 6634, 6835, 7216, 7374, 7818, 8024, 8304, 9312 (for k <= 10K) (all at n=1K)  | 1511 (999)

1674 (863)

5269 (831)

4490 (774)

6969 (764)

2014 (758)

6105 (658)

7285 (645)

3149 (627)

7669 (616)

67  | 26  | 3, 7, 31  |  | 1, 17, 21 (k = 1 at n=524K, other k at n=10K)  | 6 (4532)

11 (209)

12 (135)

7 (135)

19 (21)

5 (6)

2 (6)

22 (3)

16 (3)

25 (2)

68  | 22  | 3, 23  |  | 1, 17 (k = 1 at n=16.77M, k = 17 at n=1M)  | 12 (656921)

11 (3947)

8 (319)

16 (36)

5 (29)

13 (26)

19 (6)

10 (6)

4 (6)

18 (2)

69  | 6  | 5, 7  |  | none - proven  | 3 (2)

1 (2)

5 (1)

4 (1)

2 (1)

70  | 11077  | 13, 29, 71  |  | 70, 89, 178, 212, 283, 285, 434, 545, 581, 629, 881, 1300, 1373, 1436, 1490, 1559, 1565, 1694, 1871, 1916, 1946, 1955, 2129, 2176, 2351, 2354, 2379, 2419, 2705, 2756, 3154, 3317, 3329, 3336, 3362, 3407, 3452, 3530, 3647, 3762, 3764, 3929, 3944, 4025, 4061, 4119, 4166, 4188, 4193, 4250, 4331, 4351, 4454, 4913, 5145, 5169, 5204, 5231, 5348, 5429, 5540, 5594, 5608, 5609, 5798, 5857, 5894, 5953, 5975, 6133, 6167, 6218, 6410, 6518, 6530, 6582, 6743, 7145, 7325, 7365, 7552, 7578, 7691, 7736, 7811, 7907, 7974, 7994, 8003, 8015, 8045, 8153, 8159, 8201, 8234, 8306, 8348, 8351, 8377, 8406, 8423, 8465, 8477, 8637, 8907, 8945, 9231, 9268, 9323, 9428, 9471, 9515, 9586, 9693, 9712, 9751, 9758, 10009, 10051, 10089, 10193, 10271, 10291, 10399, 10438, 10544, 10574, 10718, 10997, 11003 (all at n=1K)  | 3479 (998)

7345 (994)

10793 (976)

4155 (970)

1040 (965)

4343 (936)

2471 (936)

5578 (932)

4208 (926)

2877 (907)

71  | 5  | 2, 3  |  | none - proven  | 4 (22)

2 (3)

1 (2)

3 (1)

72  | 731  | 5, 61, 73  |  | 72 (16.77M)  | 493 (480933)

647 (60536)

489 (20201)

559 (9626)

395 (8171)

444 (6071)

499 (2998)

292 (2779)

649 (2658)

521 (1208)

73  | 47  | 2, 5, 13  |  | none - proven (with probable primes that have not been certified: k = 14) (primality certificate for k=21, primality certificate for k=39)  | 14 (21369)

21 (1531)

39 (350)

16 (40)

8 (28)

13 (23)

25 (10)

17 (9)

36 (7)

38 (6)

74  | 4  | 3, 5  |  | none - proven  | 1 (2)

3 (1)

2 (1)

75  | 37  | 2, 19  |  | none - proven (primality certificate for k=11)  | 11 (3071)

28 (129)

17 (128)

18 (57)

12 (57)

5 (48)

1 (32)

33 (18)

35 (11)

9 (6)

76  | 34  | 7, 11  |  | none - proven  | 29 (84)

22 (16)

1 (16)

23 (12)

19 (6)

15 (6)

33 (4)

8 (4)

20 (3)

13 (3)

77  | 7  | 2, 3  |  | 1 (524K)  | 4 (6098)

2 (3)

3 (2)

6 (1)

5 (1)

78  | 96144  | 5, 79, 1217  |  | 78, 1143, 2371, 3317, 3513, 4346, 4820, 4897, 5136, 5294, 5531, 5686, 5862, 6103, 6353, 6859, 7188, 7594, 8373, 9558, 9652, 9694, 9701, 9953, 10348, 10723, 11003, 11219, 12244, 12251, 13353, 13508, 13768, 14566, 14832, 15126, 15777, 15899, 16071, 16273, 16591, 17588, 17761, 18248, 18776, 19501, 19828, 19931, 20146, 20206, 20754, 21171, 21284, 21453, 21489, 21884, 21972, 22279, 22662, 23337, 23341, 23953, 24254, 24672, 24877, 24886, 24912, 25044, 25171, 25199, 26069, 26212, 26515, 26592, 27059, 27124, 27537, 27663, 28202, 28423, 28518, 28597, 29303, 29322, 29497, 29784, 30572, 30967, 31030, 32073, 32633, 33094, 33193, 33318, 33732, 34208, 34522, 34528, 34712, 34998, 35244, 35433, 35628, 35709, 36014, 36497, 37068, 37456, 37773, 37795, 37842, 38009, 38393, 38401, 39724, 40361, 40844, 41239, 41271, 41634, 42671, 43214, 43493, 43609, 43693, 43770, 44428, 44631, 45268, 45345, 45352, 45582, 45584, 45779, 46213, 46374, 46927, 47053, 48012, 48113, 48173, 48187, 48824, 49139, 49149, 49482, 50441, 51148, 51428, 51501, 51981, 52238, 52541, 52744, 53503, 53703, 53721, 54263, 54273, 54438, 54669, 54942, 55026, 56091, 56199, 57276, 57303, 57694, 58409, 58582, 59373, 59611, 60513, 60906, 60987, 61417, 61648, 61777, 62033, 62567, 62663, 62964, 63596, 63666, 64542, 64712, 65253, 65727, 65887, 67070, 67591, 67941, 68011, 68053, 68697, 69173, 70943, 70982, 71168, 71203, 71609, 71730, 71952, 72225, 73943, 74051, 74249, 74367, 74733, 75019, 75492, 76394, 77182, 77209, 77573, 77972, 78826, 79001, 79127, 79749, 79949, 80046, 80263, 80343, 80737, 80739, 80897, 81731, 81864, 82556, 83419, 83502, 83978, 84013, 84818, 85133, 85714, 86267, 86281, 86371, 86503, 86687, 87016, 87156, 87328, 87559, 87614, 87691, 87821, 88321, 88479, 88619, 89039, 89214, 89352, 89429, 89836, 90481, 91009, 91125, 91496, 92826, 93587, 93624, 93722, 93774, 93873, 93981, 94114, 94758, 95354, 95670 (k = 78 at n=16.77M, k = 6 mod 7 and k = 10 mod 11 at n=1K, other k at n=100K)  | 31738 (98568)

83107 (95785)

25281 (83932)

22344 (83678)

12325 (83516)

78211 (82952)

74928 (80731)

34346 (78373)

60908 (70199)

46424 (66623)

79  | 9  | 2, 5  |  | none - proven (for the k=3 prime, factor N+1 is equivalent to factor 79^875+1) (for the k=5 prime, factor N-1 is equivalent to factor 79^162-1)  | 3 (875)

5 (162)

6 (2)

1 (2)

8 (1)

7 (1)

4 (1)

2 (1)

80  | 1039  | 3, 7, 13, 43, 173  |  | 86, 92, 166, 370, 393, 472, 556, 623, 692, 778, 818, 947, 968 (k = 947 at n=4K, other k at n=500K)  | 628 (491322)

295 (404886)

326 (398799) 188 (142291)

433 (121106)

770 (107149)

857 (106007)

787 (48156)

1024 (46306)

233 (36917)

81  | 575  | 2, 41  | All k=4*q^4 for all n:

let k=4*q^4

and let m=q*3^n; factors to:

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

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

| 239, 335, 514 (all at n=5K)  | 558 (51992)

311 (7834)

75 (3309)

569 (2937)

439 (2097)

284 (1455)

41 (1223)

389 (871)

34 (734)

317 (518)

| k = 4, 64, and 324 proven composite by full algebraic factors.
82  | 19587  | 5, 7, 13, 37, 83  |  | 74, 122, 167, 470, 839, 848, 1121, 1226, 1251, 1319, 1327, 1376, 1427, 1433, 1493, 1514, 1559, 1716, 1733, 1798, 1908, 2024, 2066, 2159, 2251, 2339, 2352, 2461, 2491, 2708, 2939, 2989, 3041, 3236, 3239, 3332, 3377, 3474, 3572, 3593, 3641, 3656, 3746, 3896, 3962, 4133, 4142, 4151, 4232, 4379, 4384, 4454, 4542, 4898, 5064, 5251, 5279, 5396, 5477, 5483, 5516, 5612, 5703, 5721, 5747, 5867, 5893, 5975, 6059, 6226, 6497, 6641, 6761, 6764, 6912, 6953, 7127, 7160, 7201, 7266, 7541, 7718, 7856, 7884, 7969, 7982, 8135, 8301, 8384, 8467, 8532, 8609, 8657, 8742, 8797, 8909, 9038, 9169, 9335, 9380, 9419, 9437, 9461, 9476, 9638, 9776, 9788, 9812, 9836, 9842, 9851, 9911, 9941, 9954, 10049, 10127, 10154, 10304, 10448, 10553, 10577, 10586, 10802, 10958, 11080, 11087, 11177, 11408, 11612, 11621, 11666, 11702, 11704, 11761, 11783, 11834, 11957, 11963, 11984, 12008, 12036, 12119, 12347, 12451, 12491, 12532, 12548, 12554, 12638, 12737, 12744, 12856, 12866, 12938, 12947, 12949, 13121, 13246, 13268, 13283, 13343, 13607, 13613, 13777, 14192, 14473, 14609, 14621, 14639, 14676, 14681, 14692, 14873, 14941, 14984, 15032, 15122, 15146, 15203, 15271, 15296, 15356, 15551, 15854, 15869, 15937, 15953, 16088, 16133, 16267, 16269, 16423, 16433, 16442, 16502, 16601, 16682, 16733, 16811, 16847, 17029, 17078, 17112, 17174, 17177, 17369, 17393, 17798, 17813, 17846, 17921, 18332, 18342, 18457, 18548, 18566, 18626, 18944, 18965, 18971, 19061, 19181, 19421 (k = 2 mod 3 at n=1K, other k at n=100K)  | 5652 (96054)

7288 (94205)

5101 (88245)

5977 (85004)

9676 (84109)

17692 (82887)

17091 (82407)

19134 (82154)

18168 (71000)

19098 (69654)

83  | 5  | 2, 3  |  | 1, 3 (k = 1 at n=524K, k = 3 at n=8K)  | 4 (5870)

2 (1)

84  | 16  | 5, 17  |  | none - proven  | 14 (47)

15 (6)

10 (5)

2 (4)

11 (2)

7 (2)

6 (2)

3 (2)

1 (2)

13 (1)

85  | 87  | 2, 43  |  | none - proven  | 70 (1586)

65 (125)

43 (62)

20 (57)

68 (12)

37 (12)

38 (11)

73 (7)

34 (7)

83 (6)

86  | 28  | 3, 29  |  | 1, 8 (k = 1 at n=16.77M, k = 8 at n=1M)  | 6 (40)

24 (23)

17 (17)

7 (12)

19 (6)

4 (6)

27 (4)

25 (2)

22 (2)

21 (2)

87  | 21  | 2, 11  |  | none - proven  | 12 (1214)

8 (112)

17 (16)

1 (16)

7 (7)

5 (6)

16 (4)

10 (3)

14 (2)

13 (2)

88  | 26  | 3, 7, 19, 31  |  | none - proven (primality certificate for k=8)  | 8 (1094)

14 (83)

12 (9)

6 (7)

3 (4)

23 (3)

21 (3)

11 (3)

25 (2)

22 (2)

89  | 4  | 3, 5  |  | 1 (524K)  | 3 (1)

2 (1)

90  | 27  | 7, 13  |  | none - proven  | 14 (14)

8 (14)

22 (6)

19 (6)

5 (6)

16 (4)

12 (3)

23 (2)

21 (2)

15 (2)

91  | 45  | 2, 23  |  | 1 (524K)  | 33 (52)

35 (45)

9 (36)

7 (17)

37 (12)

36 (9)

29 (8)

43 (7)

41 (6)

16 (6)

92  | 32  | 3, 31  |  | 1 (16.77M)  | 31 (416)

25 (308)

8 (109)

17 (59)

29 (47)

24 (38)

10 (24)

16 (12)

7 (6)

23 (5)

93  | 95  | 2, 47  |  | 62, 67, 87, 93 (k = 62 at n=100K, k = 93 and n=524K, other k at n=8K)  | 19 (4362)

36 (3936)

43 (2994)

31 (527)

6 (520)

3 (156)

79 (69)

71 (41)

63 (31)

18 (24)

94  | 39  | 5, 19  |  | none - proven (primality certificate for k=17)  | 17 (581)

9 (263)

11 (90)

31 (54)

2 (51)

16 (26)

23 (22)

34 (19)

30 (12)

38 (11)

95  | 5  | 2, 3  |  | none - proven  | 3 (9)

4 (6)

1 (2)

2 (1)

96  | 68869  | 13, 97, 709  |  | 194, 939, 969, 994, 1169, 1177, 1262, 1514, 1844, 2146, 2424, 2545, 2868, 2952, 3028, 3364, 3624, 3699, 3784, 4019, 4164, 4239, 4549, 5140, 5239, 5262, 5764, 5959, 6009, 6074, 6304, 6389, 6569, 6668, 6671, 6769, 6882, 6934, 7132, 7246, 7312, 7539, 7569, 8009, 8069, 8226, 8634, 8796, 9020, 9064, 9309, 9489, 9589, 9619, 9799, 10089, 10139, 10574, 10669, 10739, 10844, 10849, 10939, 11154, 11159, 11361, 11549, 11634, 11659, 11738, 11974, 12029, 12054, 12417, 12706, 12999, 13044, 13519, 13773, 13899, 14169, 14279, 14299, 14494, 14646, 15194, 15208, 15228, 15448, 16073, 16279, 16349, 16799, 17009, 17029, 17264, 17362, 17517, 17564, 17909, 18189, 18231, 18254, 18916, 19109, 19254, 19289, 19304, 19683, 19884, 19934, 20064, 20324, 20369, 20494, 20584, 20599, 20733, 21194, 21234, 21679, 22309, 22419, 22569, 22604, 22699, 22999, 23174, 23629, 24015, 24049, 24259, 24490, 24724, 25459, 25575, 25829, 25995, 26229, 26379, 26424, 26577, 26846, 26899, 26941, 27219, 27299, 27334, 27514, 27644, 27682, 27849, 28939, 29039, 29278, 29411, 29574, 30360, 30459, 30484, 30509, 30689, 30779, 31461, 31621, 31979, 32138, 32239, 32300, 32319, 32369, 32384, 32432, 32609, 32664, 32714, 33034, 33175, 33229, 34119, 34267, 34469, 34744, 35071, 35296, 35309, 35404, 35794, 36304, 36824, 36834, 37129, 37829, 38134, 38219, 38546, 38609, 38739, 39164, 39187, 39309, 39386, 39719, 39777, 39983, 40014, 40724, 41339, 41614, 41674, 41709, 41779, 41806, 41905, 42004, 42179, 42199, 42291, 42374, 42394, 42444, 42629, 42901, 42954, 42979, 43194, 43389, 43494, 43739, 43909, 43914, 44136, 44384, 44539, 44611, 44634, 45009, 45589, 45774, 46134, 46214, 46344, 46409, 46444, 46658, 46684, 47139, 47143, 47164, 47238, 47259, 47344, 47644, 48010, 48214, 48307, 48404, 48479, 48504, 48582, 48744, 48749, 48914, 49017, 49249, 49859, 50079, 50194, 50224, 50387, 50549, 50709, 50929, 51099, 51159, 51399, 51414, 51797, 51827, 52019, 52034, 52209, 53004, 53079, 53465, 53519, 53624, 54016, 54254, 54509, 54994, 55049, 55774, 55959, 56044, 56229, 56719, 56854, 56919, 56939, 57037, 57114, 57264, 57520, 57524, 57968, 58199, 58215, 58356, 58644, 59189, 59519, 59654, 59684, 59799, 59945, 59947, 60014, 60194, 60269, 60464, 60624, 60917, 61014, 61034, 61384, 61524, 61699, 61773, 62024, 62774, 62884, 62954, 63029, 63439, 63504, 63509, 63799, 63809, 63939, 64454, 64484, 64644, 64700, 64789, 64871, 64982, 65019, 65089, 65164, 65229, 65239, 65379, 65399, 65573, 65606, 65668, 65749, 65864, 66039, 66096, 66119, 66349, 66559, 66664, 66734, 66749, 66929, 67159, 67174, 67373, 67976, 68004, 68169, 68192, 68274, 68339, 68384, 68444, 68532, 68752, 68774 (k = 4 mod 5 and k = 18 mod 19 at n=1K, other k at n=100K)  | 14825 (91707)

64312 (89580)

59132 (85620)

41452 (85565)

32762 (81344)

21533 (81235)

26773 (74392)

13872 (73620)

4461 (73443)

16780 (72065)

97  | 127  | 2, 7  |  | 1, 27, 43, 62, 83, 116, 120, 123 (k = 1 at n=524K, k = 120 at n=100K, other k at n=2K)  | 64 (7474)

22 (2182)

122 (660)

68 (593)

26 (224)

87 (167)

24 (158)

113 (104)

41 (89)

17 (64)

98  | 10  | 3, 11  |  | 1 (16.77M)  | 4 (294)

8 (119)

6 (32)

7 (8)

3 (2)

9 (1)

5 (1)

2 (1)

99  | 9  | 2, 5  |  | 1 (524K)  | 5 (14)

8 (10)

6 (6)

7 (1)

4 (1)

3 (1)

2 (1)

100  | 62  | 3, 7, 13  |  | none - proven  | 31 (168)

38 (29)

59 (24)

34 (13)

36 (8)

17 (6)

52 (5)

3 (5)

60 (4)

46 (4)

101  | 7  | 2, 3  |  | none - proven  | 2 (192275)

3 (22)

5 (3)

4 (2)

1 (2)

6 (1)

102  | 293  | 7, 19, 79  |  | 122, 178, 236 (all at n=360K)  | 46 (50451)

278 (10941)

94 (6421)

12 (2739)

73 (2040)

131 (1112)

202 (610)

56 (499)

48 (305)

271 (300)

103  | 25  | 2, 13  |  | 7 (8K)  | 13 (7010)

20 (476)

11 (81)

23 (51)

14 (34)

21 (16)

5 (16)

2 (8)

8 (7)

1 (4)

104  | 4  | 3, 5  |  | 1 (16.77M)  | 2 (1233)

3 (1)

105  | 319  | 2, 53  |  | none - proven (primality certificate for k=191, primality certificate for k=39, primality certificate for k=183)  | 191 (5045)

36 (675)

39 (348)

264 (275)

183 (210)

150 (193)

80 (177)

164 (146)

167 (140)

204 (105)

106  | 2387  | 3, 19, 199  |  | 69, 110, 164, 259, 412, 449, 635, 748, 812, 929, 1088, 1190, 1429, 1511, 1607, 1628, 1823, 1925, 1985, 2018, 2075, 2177, 2189, 2216, 2279 (all at n=2K)  | 1559 (1975)

436 (1949)

679 (1818)

198 (1699)

2119 (1685)

1160 (1564)

2036 (1312)

887 (1307)

1703 (1305)

1835 (1303)

107  | 5  | 2, 3  |  | 1 (524K)  | 4 (32586)

3 (165)

2 (3)

108  | 26270  | 7, 13, 109, 127  |  | 108, 127, 156, 211, 217, 653, 998, 1267, 1271, 1854, 2252, 2393, 2399, 2724, 2842, 2915, 2942, 2976, 3098, 3563, 3571, 3925, 3938, 4162, 4311, 4391, 4468, 4623, 4699, 5013, 5117, 5251, 5778, 5794, 5849, 5924, 5994, 6686, 7211, 7478, 8401, 8623, 8642, 8828, 9127, 9482, 9578, 9941, 10188, 10202, 10245, 10574, 10689, 10973, 11008, 11028, 11321, 11335, 11703, 11833, 11909, 12172, 12209, 12427, 12534, 13081, 13299, 13316, 13844, 13861, 14044, 14134, 14691, 14932, 15207, 15638, 15912, 15913, 15926, 16042, 16122, 16240, 16569, 16896, 17267, 17616, 18319, 18638, 19098, 19158, 19294, 19318, 19839, 19948, 19966, 20303, 20687, 20929, 21181, 21262, 21511, 21532, 21581, 21818, 21908, 22008, 22182, 22194, 22259, 22266, 22562, 22706, 23066, 23327, 23543, 23838, 24078, 24088, 24103, 24529, 24756, 24767, 24853, 25062, 25068, 25071, 25319, 25546, 25607, 25763, 25973, 26234, 26256 (k = 108 at n=16.77M, other k at n=100K)  | 7612 (99261)

7304 (94930)

15874 (94153)

8034 (93577)

2874 (91402)

20666 (91335)

7631 (90728)

9187 (90213)

6759 (89530)

21101 (88027)

109  | 19  | 2, 5  |  | 1 (524K)  | 3 (6)

4 (3)

18 (2)

16 (2)

12 (2)

11 (2)

6 (2)

5 (2)

17 (1)

15 (1)

110  | 38  | 3, 37  |  | none - proven  | 20 (933)

34 (356)

11 (161)

13 (124)

19 (66)

25 (58)

2 (51)

22 (42)

28 (12)

18 (11)

111  | 13  | 2, 7  |  | none - proven  | 8 (62)

1 (16)

9 (8)

11 (5)

6 (3)

12 (2)

5 (2)

10 (1)

7 (1)

4 (1)

112  | 2261  | 5, 13, 113  |  | 209, 269, 467, 941, 1292, 1412, 1463, 1499, 1517, 1604, 1613, 1664, 1696, 1937 (k = 1696 at n=1M, other k at n=6.9K)  | 1780 (62794)

547 (8124)

953 (6802)

677 (5723)

1920 (5333)

2082 (5308)

1712 (4836)

813 (4616)

8 (4526)

1217 (3872)

113  | 20  | 3, 19  |  | 17 (8K)  | 4 (2958)

13 (1336)

19 (50)

18 (47)

8 (47)

16 (40)

12 (4)

3 (4)

1 (4)

15 (2)

114  | 24  | 5, 23  |  | none - proven  | 1 (32)

12 (15)

3 (12)

22 (11)

11 (10)

9 (5)

16 (4)

23 (3)

19 (3)

15 (3)

115  | 57  | 2, 29  |  | 17, 47 (both at n=8K)  | 30 (47376)

50 (798)

38 (94)

46 (79)

23 (51)

5 (44)

53 (38)

40 (38)

49 (14)

37 (12)

116  | 14  | 3, 13  |  | none - proven  | 12 (47)

9 (8)

4 (6)

10 (4)

7 (4)

5 (3)

13 (2)

6 (2)

1 (2)

11 (1)

117  | 119  | 2, 59  |  | 59, 117 (k = 59 at n=8K, k = 117 at n=524K)  | 58 (460033)

75 (1428)

11 (1164)

77 (311)

2 (286)

81 (264)

47 (227)

67 (182)

4 (101)

51 (76)

118  | 50  | 7, 17  |  | 48 (740K)  | 43 (106)

36 (96)

18 (80)

33 (67)

3 (46)

15 (22)

29 (10)

21 (7)

35 (6)

46 (5)

119  | 4  | 3, 5  |  | none - proven  | 1 (4)

3 (1)

2 (1)

120  | 374876369  | 11, 13, 1117, 14281  |  | 56, 89, 208, 219, 307, 309, 426, 540, 560, 694, 714, 727, 991, 1024, 1167, 1616, 1658, 1662, 1689, 1833, 1946, 1969, 1970, 2023, 2078, 2157, 2223, 2279, 2377, 2395, 2509, 2519, 2881, 3161, 3257, 3301, 3321, 3345, 3387, 3510, 3561, 3598, 3607, 3774, 3805, 3814, 3827, 3860, 3893, 3950, 4212, 4333, 4367, 4456, 4610, 4724, 4762, 4852, 4993, 5069, 5191, 5347, 5433, 5543, 5553, 5665, 5763, 5875, 5894, 5928, 6029, 6084, 6447, 6478, 6502, 6715, 6718, 6984, 7097, 7195, 7248, 7284, 7379, 7589, 7998, 8051, 8161, 8189, 8293, 8304, 8359, 8382, 8427, 8514, 8636, 8669, 8678, 8693, 8876, 8931, 8933, 8957, 9041, 9043, 9058, 9109, 9140, 9195, 9318, 9351, 9494, 9513, 9637, 9721, 9890 (for k <= 10K) (all at n=1K)  | 8389 (969)

6546 (954)

3195 (951)

3466 (908)

7479 (899)

3359 (897)

4437 (870)

8584 (843)

6382 (803)

738 (790)

121  | 27  | 7, 19, 37  |  | none - proven  | 23 (102)

24 (72)

7 (6)

17 (5)

10 (5)

2 (5)

25 (4)

21 (4)

19 (4)

16 (4)

122  | 40  | 3, 41  |  | 1, 34 (k = 1 at n=16.77M, k = 34 at n=1M)  | 37 (1622)

31 (1236)

16 (764)

2 (755)

25 (674)

23 (389)

17 (371)

4 (358)

5 (135)

28 (108)

123  | 55  | 2, 17, 89  |  | 1, 3, 41 (k = 1 at n=524K, other k at n=8K)  | 19 (59)

38 (42)

47 (29)

13 (28)

34 (19)

28 (19)

8 (16)

54 (15)

15 (15)

53 (14)

124  | 31001  | 3, 5, 7, 5167  |  | 54, 61, 76, 83, 89, 96, 114, 121, 146, 171, 206, 209, 221, 239, 251, 344, 362, 376, 381, 386, 411, 416, 431, 446, 449, 516, 519, 526, 530, 576, 581, 601, 635, 646, 647, 656, 661, 669, 670, 676, 684, 731, 766, 794, 804, 809, 831, 836, 841, 872, 896, 911, 971, 976, 1019, 1031, 1051, 1054, 1076, 1111, 1124, 1129, 1136, 1166, 1190, 1229, 1251, 1254, 1259, 1264, 1284, 1298, 1324, 1326, 1336, 1369, 1421, 1446, 1460, 1461, 1471, 1474, 1477, 1499, 1519, 1535, 1536, 1551, 1569, 1586, 1591, 1601, 1604, 1647, 1657, 1676, 1686, 1700, 1721, 1727, 1734, 1741, 1779, 1801, 1814, 1829, 1844, 1864, 1910, 1955, 2021, 2034, 2036, 2045, 2055, 2067, 2069, 2096, 2097, 2109, 2114, 2129, 2159, 2163, 2179, 2216, 2234, 2266, 2306, 2316, 2354, 2374, 2375, 2406, 2414, 2429, 2436, 2446, 2462, 2504, 2507, 2539, 2559, 2561, 2565, 2621, 2639, 2646, 2651, 2716, 2726, 2734, 2799, 2821, 2834, 2840, 2844, 2861, 2864, 2874, 2901, 2906, 2934, 2981, 2999, 3019, 3032, 3041, 3049, 3053, 3071, 3144, 3161, 3164, 3181, 3229, 3236, 3242, 3251, 3281, 3285, 3296, 3299, 3316, 3329, 3351, 3405, 3442, 3470, 3471, 3491, 3494, 3533, 3554, 3561, 3574, 3631, 3659, 3674, 3684, 3714, 3726, 3736, 3737, 3758, 3779, 3806, 3824, 3854, 3869, 3881, 3890, 3911, 3916, 3921, 3941, 3961, 3981, 3986, 3994, 4021, 4049, 4086, 4089, 4124, 4127, 4131, 4153, 4162, 4191, 4196, 4226, 4231, 4254, 4297, 4306, 4314, 4352, 4375, 4388, 4406, 4414, 4421, 4454, 4476, 4489, 4500, 4506, 4520, 4521, 4529, 4541, 4546, 4589, 4594, 4604, 4629, 4719, 4739, 4751, 4764, 4769, 4799, 4849, 4891, 4910, 4926, 4936, 4952, 4961, 4964, 4973, 4974, 5001, 5041, 5048, 5049, 5108, 5114, 5121, 5149, 5154, 5189, 5191, 5231, 5244, 5279, 5289, 5300, 5316, 5321, 5326, 5364, 5366, 5369, 5375, 5381, 5384, 5414, 5440, 5462, 5474, 5481, 5489, 5519, 5543, 5551, 5579, 5581, 5596, 5651, 5663, 5681, 5696, 5697, 5701, 5721, 5723, 5738, 5744, 5771, 5781, 5799, 5801, 5816, 5819, 5825, 5839, 5840, 5851, 5876, 5884, 5909, 5919, 5939, 5951, 5976, 5981, 6024, 6026, 6036, 6041, 6046, 6059, 6099, 6146, 6151, 6161, 6164, 6166, 6196, 6201, 6211, 6219, 6224, 6241, 6269, 6296, 6310, 6323, 6329, 6366, 6383, 6386, 6394, 6401, 6409, 6410, 6411, 6416, 6486, 6494, 6496, 6511, 6514, 6536, 6539, 6559, 6596, 6620, 6621, 6644, 6646, 6647, 6654, 6659, 6665, 6686, 6689, 6691, 6712, 6729, 6731, 6746, 6749, 6751, 6761, 6789, 6794, 6806, 6821, 6864, 6881, 6891, 6904, 6908, 6926, 6949, 6956, 6959, 6962, 6971, 7004, 7006, 7016, 7034, 7036, 7071, 7074, 7079, 7081, 7146, 7169, 7204, 7216, 7227, 7239, 7259, 7269, 7271, 7276, 7301, 7319, 7324, 7331, 7359, 7369, 7376, 7391, 7424, 7439, 7446, 7451, 7454, 7472, 7484, 7486, 7499, 7523, 7544, 7559, 7565, 7586, 7601, 7609, 7639, 7656, 7664, 7666, 7671, 7691, 7739, 7744, 7761, 7796, 7801, 7831, 7851, 7868, 7881, 7886, 7931, 7949, 7979, 7981, 8014, 8017, 8034, 8042, 8054, 8114, 8141, 8146, 8192, 8213, 8219, 8221, 8231, 8274, 8279, 8291, 8296, 8321, 8323, 8351, 8354, 8381, 8396, 8417, 8423, 8424, 8429, 8516, 8519, 8526, 8531, 8532, 8579, 8634, 8641, 8651, 8666, 8681, 8711, 8714, 8741, 8771, 8776, 8780, 8786, 8829, 8831, 8876, 8916, 8921, 8930, 8936, 8939, 8966, 8978, 8982, 9006, 9024, 9026, 9038, 9069, 9099, 9106, 9118, 9138, 9161, 9166, 9173, 9187, 9209, 9214, 9216, 9226, 9244, 9261, 9267, 9269, 9286, 9302, 9314, 9319, 9411, 9479, 9483, 9509, 9521, 9536, 9594, 9596, 9598, 9599, 9641, 9651, 9681, 9687, 9743, 9754, 9785, 9791, 9831, 9836, 9865, 9866, 9901, 9911, 9914, 9949, 9971 (for k <= 10K) (all at n=1K)  | 1646 (998)

8094 (997)

1886 (996)

1926 (994)

2987 (985)

7193 (981)

3276 (974)

6974 (973)

6951 (966)

2801 (960)

125  | 7  | 2, 3  | All k = m^3 for all n;

factors to:

(m*5^n + 1) *

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

| none - proven  | 4 (2)

3 (2)

6 (1)

5 (1)

2 (1)

| k = 1 proven composite by full algebraic factors.
126  | 766700  | 13, 19, 127, 829  |  | 259, 1084, 1117, 1154, 2708, 2922, 3735, 3982, 5093, 5099, 5392, 5529, 5587, 6059, 6478, 6772, 7817, 8150, 8304, 8659, 8759, 8779, 8829, 9268, 9429, 9474, 9624, 10072, 10540, 11008, 11429, 12094, 12414, 12750, 12757, 12799, 12900, 13111, 13129, 13264, 13274, 13309, 14299, 14390, 14538, 14598, 15402, 15454, 15781, 15876, 15883, 16312, 17300, 18119, 18394, 18594, 18795, 19421, 19479, 19484, 19499, 19559, 19894, 20326, 20394, 20609, 20914, 21083, 21369, 21679, 21694, 21999, 22582, 24023, 24119, 24543, 24764, 25399, 25624, 25739, 25757, 25913, 26374, 26441, 27179, 27884, 27948, 28222, 28374, 28602, 28729, 29590 (for k <= 30K) (k = 4 mod 5 at n=1K, other k at n=25K)  | 26532 (23264)

27765 (22565)

15493 (22097)

25722 (20095)

29405 (19897)

28188 (17368)

25575 (17359)

26036 (15264)

27433 (14598)

12965 (14155)

127  | 6343  | 2, 5, 17, 137  |  | 1, 37, 67, 103, 121, 134, 138, 139, 141, 153, 172, 177, 189, 201, 205, 215, 223, 237, 247, 263, 267, 301, 311, 343, 367, 381, 383, 387, 398, 409, 413, 425, 447, 452, 465, 469, 474, 487, 495, 525, 527, 529, 543, 569, 582, 601, 629, 645, 647, 649, 657, 659, 673, 681, 691, 701, 707, 727, 733, 763, 781, 790, 797, 807, 809, 818, 819, 837, 847, 849, 887, 895, 901, 903, 907, 909, 925, 927, 941, 954, 1011, 1021, 1023, 1043, 1075, 1079, 1103, 1109, 1121, 1123, 1147, 1161, 1165, 1167, 1169, 1173, 1193, 1199, 1201, 1229, 1232, 1237, 1239, 1243, 1244, 1261, 1303, 1309, 1322, 1329, 1343, 1351, 1357, 1362, 1379, 1381, 1383, 1403, 1417, 1423, 1425, 1427, 1431, 1439, 1441, 1461, 1463, 1466, 1472, 1483, 1487, 1494, 1515, 1543, 1544, 1547, 1549, 1553, 1557, 1565, 1574, 1581, 1583, 1603, 1607, 1615, 1621, 1641, 1649, 1686, 1691, 1719, 1723, 1741, 1742, 1747, 1753, 1754, 1765, 1783, 1785, 1793, 1801, 1808, 1815, 1827, 1841, 1849, 1861, 1875, 1887, 1917, 1921, 1954, 1961, 1981, 1987, 1997, 2001, 2022, 2027, 2041, 2055, 2083, 2089, 2109, 2123, 2147, 2152, 2156, 2167, 2177, 2181, 2189, 2211, 2229, 2235, 2241, 2261, 2263, 2265, 2285, 2287, 2330, 2335, 2336, 2341, 2375, 2401, 2403, 2409, 2429, 2441, 2461, 2521, 2523, 2531, 2537, 2551, 2603, 2607, 2625, 2627, 2636, 2649, 2657, 2661, 2687, 2701, 2721, 2729, 2741, 2744, 2749, 2778, 2801, 2803, 2809, 2847, 2861, 2863, 2867, 2869, 2887, 2894, 2907, 2908, 2909, 2915, 2921, 2929, 2949, 2961, 2963, 2977, 2981, 2987, 2988, 2993, 3001, 3005, 3041, 3045, 3061, 3069, 3089, 3093, 3095, 3099, 3107, 3121, 3129, 3133, 3141, 3143, 3169, 3181, 3199, 3209, 3221, 3241, 3243, 3276, 3283, 3297, 3303, 3309, 3313, 3325, 3327, 3329, 3345, 3363, 3377, 3381, 3392, 3401, 3407, 3419, 3421, 3449, 3455, 3461, 3489, 3501, 3521, 3526, 3527, 3533, 3543, 3545, 3549, 3563, 3603, 3641, 3646, 3647, 3703, 3741, 3743, 3747, 3763, 3779, 3790, 3807, 3811, 3812, 3815, 3821, 3823, 3829, 3896, 3923, 3929, 3947, 3981, 3986, 3987, 3995, 3996, 4001, 4007, 4021, 4029, 4031, 4039, 4045, 4063, 4073, 4079, 4081, 4087, 4112, 4125, 4135, 4157, 4164, 4167, 4181, 4185, 4193, 4201, 4207, 4229, 4241, 4247, 4261, 4281, 4289, 4309, 4323, 4327, 4329, 4339, 4364, 4373, 4381, 4382, 4385, 4416, 4421, 4437, 4447, 4455, 4469, 4481, 4503, 4517, 4521, 4527, 4531, 4547, 4573, 4587, 4609, 4614, 4617, 4643, 4645, 4667, 4677, 4684, 4701, 4705, 4742, 4761, 4781, 4809, 4819, 4823, 4829, 4849, 4867, 4887, 4891, 4896, 4909, 4957, 4968, 4969, 4975, 4987, 4995, 5005, 5009, 5016, 5023, 5025, 5041, 5057, 5061, 5067, 5069, 5091, 5101, 5119, 5123, 5149, 5165, 5172, 5187, 5189, 5201, 5205, 5226, 5238, 5247, 5249, 5267, 5273, 5283, 5321, 5327, 5331, 5343, 5347, 5363, 5368, 5379, 5381, 5387, 5391, 5399, 5415, 5429, 5435, 5441, 5443, 5457, 5461, 5469, 5477, 5485, 5487, 5488, 5503, 5507, 5529, 5531, 5534, 5543, 5547, 5549, 5563, 5577, 5583, 5589, 5606, 5609, 5615, 5618, 5619, 5622, 5623, 5627, 5638, 5665, 5668, 5674, 5678, 5687, 5697, 5701, 5707, 5713, 5721, 5723, 5735, 5747, 5761, 5767, 5799, 5807, 5813, 5823, 5837, 5841, 5859, 5861, 5863, 5867, 5887, 5888, 5903, 5923, 5929, 5941, 5955, 5957, 5966, 5981, 5996, 6015, 6021, 6041, 6047, 6048, 6057, 6081, 6085, 6087, 6111, 6114, 6121, 6149, 6209, 6221, 6231, 6237, 6245, 6261, 6269, 6275, 6277 (all at n=1K)  | 2163 (985)

2837 (982)

6065 (980)

2479 (975)

3525 (972)

365 (968)

5541 (964)

5654 (963)

6129 (950)

2267 (947)

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)

| 16, 40 (k = 16 at n=4.908G, k = 40 at n=1.2857M)  | 41 (39271)

42 (13001)

20 (473)

28 (322)

38 (291)

19 (178)

25 (64)

3 (27)

17 (21)

31 (20)

| k = 1 proven composite by full algebraic factors.

k = 8 and 32 have no possible prime.

129  | 14  | 5, 13  |  | none - proven  | 6 (16796)

4 (19)

9 (15)

2 (6)

1 (4)

11 (2)

5 (2)

13 (1)

12 (1)

10 (1)

130  | 1049  | 3, 7, 31, 131  |  | 37, 50, 71, 227, 341, 414, 545, 794, 809, 920, 1013 (all at n=2K)  | 992 (1751)

458 (1399)

773 (1303)

593 (917)

944 (880)

83 (695)

278 (662)

272 (614)

1046 (612)

290 (543)

131  | 5  | 2, 3  |  | none - proven  | 4 (2)

1 (2)

3 (1)

2 (1)

132  | 13  | 5, 7, 17  |  | none - proven  | 6 (5)

1 (4)

7 (3)

12 (2)

9 (2)

8 (2)

4 (2)

2 (2)

11 (1)

10 (1)

133  | 59  | 2, 5, 29  |  | 23, 51 (both at n=2K)  | 19 (806)

57 (174)

38 (43)

48 (18)

43 (12)

58 (10)

45 (8)

41 (8)

27 (8)

8 (7)

134  | 4  | 3, 5  |  | none - proven  | 3 (4)

1 (2)

2 (1)

135  | 33  | 2, 17  |  | 1, 17 (k = 1 at n=524K, k = 17 at n=2K)  | 21 (1154)

7 (213)

10 (54)

25 (38)

20 (28)

32 (13)

3 (9)

28 (8)

8 (8)

5 (4)

136  | 29180  | 53, 137, 349  |  | testing not started  | testing not started  |
137  | 22  | 3, 23  |  | 1, 5, 17 (k = 1 at n=524K, other k at n=2K)  | 2 (327)

10 (102)

14 (93)

16 (48)

11 (19)

4 (18)

13 (4)

7 (4)

12 (3)

19 (2)

138  | 2781  | 5, 13, 139  |  | 138, 211, 344, 678, 1188, 1444, 1494, 1818, 2371, 2627 (k = 138 at n=16.77M, other k at n=500K)  | 2636 (469911)

2189 (345010)

2354 (314727)

1019 (274533)

1789 (271671)

141 (244616)

2416 (214921)

866 (212835)

2062 (192750)

47 (136218)

139  | 6  | 3, 5  |  | none - proven  | 5 (6)

2 (5)

3 (3)

1 (2)

4 (1)

140  | 46  | 3, 47  |  | 8 (1M)  | 16 (251178)

34 (136)

29 (103)

38 (79)

13 (64)

28 (44)

11 (37)

44 (31)

10 (24)

14 (23)

141  | 143  | 2, 71  |  | 19, 27, 64, 107 (all at n=2K)  | 123 (312)

95 (109)

7 (99)

46 (75)

129 (73)

39 (53)

77 (47)

17 (45)

15 (25)

93 (24)

142  | 12  | 11, 13  |  | none - proven  | 10 (407)

7 (23)

2 (4)

1 (4)

5 (3)

3 (2)

11 (1)

9 (1)

8 (1)

6 (1)

143  | 5  | 2, 3  |  | 1 (524K)  | 3 (183)

4 (10)

2 (5)

144  | 59  | 5, 29  |  | 1 (16.77M)  | 34 (3061)

37 (1154)

6 (782)

31 (102)

55 (88)

30 (72)

35 (42)

17 (39)

46 (16)

40 (15)

145  | 1023  | 2, 73  |  | 18, 58, 94, 220, 221, 367, 458, 539, 628, 719, 729, 783, 795, 802, 863, 904 (all at n=2K)  | 72 (769)

559 (734)

490 (632)

335 (586)

940 (512)

951 (506)

336 (448)

8 (401)

989 (397)

176 (396)

146  | 8  | 3, 7  |  | none - proven  | 5 (3)

7 (2)

4 (2)

1 (2)

6 (1)

3 (1)

2 (1)

147  | 73  | 2, 37  |  | 1, 17, 19, 35, 47, 63 (k = 1 at n=524K, other k at n=2K)  | 66 (520)

65 (434)

69 (226)

43 (201)

2 (154)

37 (152)

61 (136)

25 (128)

14 (115)

54 (62)

148  | 3128  | 5, 13, 149  |  | 43, 98, 148, 168, 246, 299, 302, 359, 392, 413, 416, 464, 563, 641, 684, 728, 768, 776, 802, 876, 941, 953, 963, 1091, 1093, 1101, 1103, 1136, 1166, 1185, 1295, 1322, 1379, 1418, 1427, 1496, 1559, 1611, 1633, 1638, 1652, 1669, 1799, 1808, 1877, 1901, 2064, 2072, 2107, 2162, 2207, 2361, 2417, 2548, 2573, 2576, 2716, 2745, 2852, 2933, 2978, 2981, 2996, 3029, 3033, 3038, 3071, 3112 (all at n=2K)  | 2369 (1947)

338 (1947)

1781 (1829)

134 (1783)

2467 (1709)

1256 (1705)

1571 (1696)

1787 (1677)

1586 (1644)

1676 (1541)

149  | 4  | 3, 5  |  | 1 (524K)  | 2 (3)

3 (2)

150  | 49074  | 7, 31, 103, 151  |  | 343, 1553, 3980, 4578, 5254, 5413, 5891, 6041, 7342, 7506, 7724, 8787, 8906, 10256, 10699, 11434, 11465, 11475, 12232, 13591, 14265, 16046, 17366, 18806, 19256, 19480, 20235, 20537, 20789, 20988, 21388, 22045, 22604, 23307, 24765, 24914, 25364, 26478, 26909, 27320, 27502, 29265, 29446, 30501, 30654, 31666, 33674, 34594, 35391, 35484, 36265, 36774, 40232, 40839, 41073, 42128, 42734, 43093, 43200, 43275, 44242, 44441, 45161, 46649, 46660, 47111, 48168, 48354, 48617 (all at n=100K)  | 2529 (95448)

25295 (93740)

43789 (91123)

30505 (91058)

15402 (88775)

610 (87338)

41663 (83930)

22810 (81558)

26349 (75650)

22237 (72247)

151  | 37  | 2, 19  |  | 1 (524K)  | 15 (925)

25 (166)

32 (63)

20 (40)

8 (19)

19 (11)

17 (10)

30 (8)

7 (7)

33 (6)

152  | 16  | 3, 17  |  | none - proven  | 11 (837)

6 (27)

4 (18)

13 (8)

1 (8)

9 (7)

12 (4)

2 (3)

10 (2)

7 (2)

153  | 15  | 2, 7  |  | none - proven  | 13 (79)

3 (4)

1 (4)

12 (2)

9 (2)

8 (2)

7 (2)

14 (1)

11 (1)

10 (1)

154  | 61  | 5, 31  |  | none - proven (for the k=16 prime, factor N-1 is equivalent to factor 154^252-1)  | 40 (9256)

16 (252)

36 (138)

44 (89)

31 (88)

37 (79)

59 (17)

43 (15)

9 (15)

26 (8)

155  | 5  | 2, 3  |  | 1, 4 (k = 1 at n=524K, k = 4 at n=1.5M)  | 3 (1)

2 (1)

156  | unknown (>10^9, <=18406311208)  | unknown  |  | testing not started  | testing not started  |
157  | 47  | 2, 5, 17  |  | 15, 17, 23 (all at n=2K)  | 18 (3873)

29 (1650)

38 (492)

44 (449)

30 (132)

35 (92)

20 (63)

46 (49)

40 (33)

41 (27)

158  | 52  | 3, 53  |  | none - proven  | 8 (123475)

48 (24191)

32 (13401)

38 (10519)

27 (4966)

20 (1633)

37 (1034)

4 (874)

43 (178)

47 (141)

159  | 9  | 2, 5  |  | none - proven (primality certificate for k=5)  | 5 (234)

4 (29)

8 (5)

2 (3)

6 (2)

1 (2)

7 (1)

3 (1)

160  | 22  | 7, 23  |  | 20 (2K)  | 18 (27)

14 (5)

16 (4)

9 (4)

8 (4)

7 (4)

6 (3)

15 (2)

12 (2)

5 (2)

161  | 95  | 2, 3  |  | 1, 47, 79 (k = 1 at n=524K, other k at n=2K)  | 5 (5627)

4 (4650)

53 (1603)

26 (57)

40 (52)

91 (48)

13 (44)

61 (40)

19 (40)

83 (39)

162  | 6193  | 5, 13, 37, 61, 163  |  | 363, 685, 916, 1248, 1438, 2358, 2603, 2609, 2757, 2841, 2874, 2953, 3002, 3096, 3562, 3856, 3961, 4297, 4409, 4654, 4831, 4871, 5039, 5102, 5242, 5706, 5869, 6002 (k = 6 mod 7 and k = 22 mod 23 at n=2K, other k at n=300K)  | 6102 (230090)

2212 (227663)

3052 (200790)

1764 (76926)

3496 (60128)

1250 (58127)

933 (55381)

2163 (49760)

2377 (47102)

1398 (33797)

163  | 81  | 2, 41  |  | 8, 12, 38, 41, 63, 73 (k = 8 at n=6K, k = 12 at n=500K, other k at n=2K)  | 66 (107651)

6 (1303)

27 (409)

17 (374)

21 (236)

23 (175)

65 (148)

69 (134)

61 (84)

53 (50)

164  | 4  | 3, 5  |  | none - proven  | 3 (4)

1 (4)

2 (3)

165  | 167  | 2, 83  |  | 43 (2K)  | 80 (1104)

143 (703)

87 (589)

131 (300)

82 (273)

34 (269)

103 (137)

23 (135)

75 (74)

13 (40)

166  | 335  | 3, 7, 13, 167  |  | 29, 137, 141, 166, 208, 209, 243, 269, 326 (all at n=2K)  | 101 (1049)

113 (318)

225 (277)

334 (156)

149 (132)

191 (129)

230 (99)

107 (86)

123 (84)

95 (81)

167  | 5  | 2, 3  |  | none - proven  | 2 (6547)

1 (16)

4 (10)

3 (1)

168  | 9244  | 5, 13, 17, 73  |  | 1, 77, 248, 298, 467, 469, 740, 818, 901, 1236, 1377, 1437, 1886, 1998, 2183, 2211, 2378, 2406, 2731, 2770, 2963, 2991, 3057, 3514, 3654, 3717, 3977, 4161, 4174, 4224, 4226, 4382, 4441, 4499, 4517, 4616, 4746, 4913, 5303, 5381, 5474, 5526, 5539, 5680, 5812, 5981, 6083, 6124, 6166, 6241, 6319, 6356, 6382, 6772, 6787, 6824, 6967, 7032, 7099, 7123, 7292, 7422, 7541, 7697, 7708, 7736, 7916, 8164, 8293, 8334, 8971, 9138 (k = 1 at n=16.77M, k = 4174 at n=2K, other k at n=100K)  | 1561 (97864)

1398 (80456)

5942 (77280)

4432 (73477)

8072 (68617)

7188 (62211)

3394 (55546)

2614 (54002)

7240 (50425)

6892 (48868)

169  | 16  | 5, 17  |  | none - proven (for the k=11 prime, factor N-1 is equivalent to factor 169^282-1)  | 11 (282)

7 (8)

14 (3)

10 (2)

8 (2)

6 (2)

5 (2)

1 (2)

15 (1)

13 (1)

170  | 20  | 3, 19  |  | none - proven  | 7 (178)

5 (175)

19 (36)

17 (21)

13 (4)

3 (3)

2 (3)

16 (2)

10 (2)

4 (2)

171  | 85  | 2, 43  |  | 23, 29, 31, 39, 45, 73 (all at n=2K)  | 30 (229506)

17 (370)

69 (212)

71 (127)

77 (98)

79 (65)

58 (36)

84 (31)

37 (18)

57 (14)

172  | 62  | 3, 7, 13  |  | none - proven (primality certificate for k=26, primality certificate for k=59)  | 26 (287)

52 (259)

59 (214)

22 (108)

17 (84)

54 (35)

51 (35)

48 (26)

40 (23)

19 (15)

173  | 7  | 2, 3  |  | none - proven  | 1 (16)

4 (10)

3 (2)

6 (1)

5 (1)

2 (1)

174  | 6  | 3, 5  |  | 4 (1M)  | 1 (4)

3 (1)

2 (1)

175  | 21  | 2, 11  |  | none - proven  | 5 (64)

15 (59)

20 (36)

11 (9)

9 (8)

14 (7)

13 (6)

18 (3)

10 (3)

2 (3)

176  | 58  | 3, 59  |  | 55 (2K)  | 32 (3591)

37 (3088)

35 (995)

50 (213)

10 (146)

49 (108)

28 (24)

46 (16)

31 (14)

27 (14)

177  | 79  | 2, 5, 13  |  | none - proven (primality certificate for k=77)  | 12 (3810)

77 (646)

8 (64)

33 (54)

41 (40)

67 (36)

24 (30)

15 (18)

48 (14)

63 (13)

178  | 569  | 3, 13, 19  |  | 32, 41, 83, 96, 126, 128, 136, 155, 167, 178, 194, 212, 217, 251, 278, 283, 284, 357, 359, 372, 382, 383, 398, 407, 458, 468, 474, 480, 506, 550, 566 (all at n=2K)  | 433 (1888)

362 (1821)

410 (1626)

488 (1248)

353 (1207)

331 (1028)

363 (1018)

8 (956)

214 (889)

442 (840)

179  | 4  | 3, 5  |  | 1 (524K)  | 3 (1)

2 (1)

180  | 1679679  | 7, 31, 181, 1051  |  | testing not started  | testing not started  |
181  | 15  | 2, 7  |  | none - proven  | 8 (10)

11 (6)

13 (5)

12 (3)

14 (2)

4 (2)

3 (2)

2 (2)

1 (2)

10 (1)

182  | 23  | 3, 5, 53  |  | 1, 8 (k = 1 at n=16.77M, k = 8 at n=1M)  | 9 (263)

19 (90)

4 (70)

2 (15)

13 (12)

20 (5)

18 (4)

16 (4)

7 (4)

17 (3)

183  | 45  | 2, 23  |  | 1, 5, 9, 41 (k = 1 at n=524K, other k at n=2K)  | 24 (298)

33 (198)

38 (112)

11 (59)

29 (58)

12 (48)

14 (46)

3 (35)

37 (32)

13 (24)

184  | 36  | 5, 37  |  | none - proven (primality certificate for k=20)  | 20 (1298)

16 (298)

23 (70)

6 (40)

4 (29)

32 (16)

3 (11)

12 (10)

29 (9)

10 (9)

185  | 23  | 2, 3  |  | 10, 22 (k = 10 at n=1M, k = 22 at n=2K)  | 19 (540)

4 (414)

6 (170)

13 (98)

1 (8)

21 (3)

17 (3)

9 (3)

2 (3)

16 (2)

186  | 67  | 11, 17  |  | 1, 34 (k = 1 at n=16.77M, k = 34 at n=2K)  | 65 (18879)

56 (300)

24 (258)

35 (134)

16 (107)

40 (98)

52 (72)

45 (58)

54 (29)

50 (25)

187  | 47  | 2, 5, 13  |  | 5, 9, 29, 47, 51, 53, 61 (all at n=2K)  | 49 (938)

23 (801)

59 (141)

27 (71)

41 (68)

31 (55)

67 (47)

65 (46)

15 (43)

50 (24)

188  | 8  | 3, 7  |  | none - proven  | 4 (26)

1 (16)

2 (9)

7 (2)

3 (2)

6 (1)

5 (1)

189  | 19  | 2, 5  |  | 1 (524K)  | 18 (171175)

16 (42)

6 (34)

8 (7)

11 (4)

3 (4)

9 (3)

10 (2)

5 (2)

17 (1)

190  | 2157728  | 13, 191, 2777  |  | testing not started  | testing not started  |
191  | 5  | 2, 3  |  | 3 (6K)  | 1 (32)

4 (6)

2 (1)

192  | 7879  | 5, 7, 13, 31, 101  |  | 712, 787, 1031, 1157, 1234, 1369, 1388, 1627, 1806, 1828, 1899, 1929, 1931, 1965, 2311, 2313, 2461, 2482, 2521, 2537, 2672, 2807, 2928, 2988, 3020, 3346, 3604, 3827, 3929, 4024, 4054, 4672, 4768, 4826, 4859, 5010, 5059, 5147, 5262, 5373, 5752, 5927, 5958, 5982, 6133, 6257, 6474, 6523, 6968, 6995, 7152, 7414, 7437, 7528, 7600, 7666, 7822 (k = 2482 at n=2K, other k at n=100K)  | 1122 (89238)

5594 (86270)

5675 (74618)

3473 (69049)

4566 (67168)

2829 (63997)

6878 (60430)

5375 (54124)

6898 (52349)

7586 (49923)

193  | 2687  | 2, 3, 5, 7, 37  |  | 5, 24, 63, 68, 98, 122, 131, 150, 167, 188, 193, 203, 264, 271, 290, 293, 299, 320, 333, 367, 371, 412, 413, 419, 486, 527, 542, 545, 586, 608, 632, 678, 680, 719, 722, 731, 733, 775, 790, 819, 821, 831, 852, 962, 971, 977, 1010, 1013, 1028, 1034, 1046, 1050, 1064, 1066, 1069, 1091, 1097, 1112, 1141, 1153, 1156, 1163, 1187, 1195, 1201, 1262, 1274, 1294, 1333, 1340, 1349, 1355, 1357, 1393, 1403, 1412, 1418, 1427, 1437, 1446, 1451, 1456, 1464, 1466, 1469, 1487, 1504, 1517, 1613, 1623, 1653, 1676, 1679, 1753, 1784, 1796, 1832, 1844, 1873, 1916, 1922, 1928, 1943, 1946, 1970, 1977, 1980, 1981, 1986, 2005, 2008, 2052, 2062, 2070, 2091, 2105, 2114, 2168, 2177, 2213, 2225, 2246, 2264, 2306, 2329, 2348, 2354, 2367, 2385, 2426, 2434, 2442, 2446, 2460, 2489, 2506, 2511, 2520, 2523, 2525, 2554, 2558, 2572, 2581, 2593, 2602, 2603, 2621, 2623 (all at n=2K)  | 2243 (1839)

292 (1830)

194 (1767)

929 (1763)

1049 (1729)

1238 (1702)

518 (1699)

956 (1673)

2643 (1635)

214 (1622)

194  | 4  | 3, 5  |  | none - proven  | 1 (4)

3 (2)

2 (1)

195  | 13  | 2, 7  |  | none - proven (the k=11 prime is proven prime by N-1, and primality certificate for the large prime factor of N-1)  | 11 (239)

8 (16)

2 (6)

9 (4)

4 (3)

5 (2)

1 (2)

12 (1)

10 (1)

7 (1)

196  | 16457  | 3, 61, 211  |  | 84, 155, 196, 208, 335, 421, 434, 481, 497, 729, 974, 1262, 1268, 1271, 1313, 1378, 1397, 1494, 1553, 1770, 1854, 1861, 1913, 1971, 2024, 2027, 2036, 2078, 2096, 2168, 2378, 2480, 2541, 2547, 2558, 2561, 2615, 2643, 2705, 2779, 2839, 2881, 2954, 3023, 3044, 3110, 3230, 3472, 3503, 3658, 3689, 3722, 3830, 3851, 3938, 4286, 4377, 4451, 4523, 4574, 4730, 4886, 4924, 4952, 5088, 5116, 5123, 5149, 5274, 5302, 5342, 5378, 5444, 5477, 5557, 5714, 5759, 5770, 5771, 5794, 5810, 5909, 6026, 6038, 6116, 6139, 6179, 6221, 6354, 6541, 6654, 6674, 6715, 6716, 6784, 6896, 6962, 7006, 7009, 7090, 7102, 7175, 7301, 7442, 7544, 7595, 7637, 7697, 7760, 7827, 7871, 7904, 8261, 8324, 8363, 8405, 8434, 8539, 8648, 8664, 8684, 8771, 8807, 8819, 8876, 8896, 9103, 9104, 9113, 9206, 9286, 9393, 9415, 9494, 9641, 9743, 9852, 9929, 10016, 10093, 10139, 10199, 10215, 10313, 10325, 10474, 10524, 10613, 10655, 10757, 10830, 10832, 10889, 10905, 10919, 10920, 10973, 10979, 11015, 11165, 11228, 11258, 11314, 11348, 11519, 11586, 11591, 11624, 11699, 11831, 11952, 11971, 12209, 12238, 12446, 12458, 12464, 12493, 12614, 12766, 12782, 12814, 12899, 12923, 12938, 13070, 13088, 13092, 13198, 13251, 13364, 13414, 13421, 13430, 13436, 13556, 13566, 13571, 13595, 13631, 13664, 13700, 13745, 13791, 13859, 13982, 14090, 14091, 14104, 14123, 14144, 14255, 14348, 14414, 14435, 14438, 14444, 14569, 14588, 14625, 14670, 14711, 14715, 14759, 14810, 14823, 14900, 14959, 14971, 15083, 15098, 15172, 15317, 15362, 15485, 15659, 15728, 15835, 15861, 16133, 16187, 16208, 16265, 16286, 16350, 16391 (all at n=2K)  | 789 (1926)

9609 (1914)

3618 (1887)

9530 (1823)

15177 (1804)

14390 (1790)

2082 (1774)

13983 (1772)

14585 (1767)

11387 (1767)

197  | 7  | 2, 3  |  | 1 (524K)  | 4 (6)

6 (5)

3 (4)

5 (3)

2 (3)

198  | 4105  | 7, 13, 19, 2053  |  | 173, 311, 374, 381, 486, 714, 907, 979, 996, 1193, 1195, 1298, 1338, 1557, 1678, 1762, 1812, 1889, 1991, 2064, 2071, 2166, 2196, 2287, 2389, 2400, 2427, 2817, 2924, 3058, 3338, 3431, 3618, 3891, 3981, 4016, 4065 (k = 2166 at n=2K, other k at n=100K)  | 1074 (86150)

2976 (78439)

4014 (73851)

2864 (62462)

2084 (56478)

706 (55247)

2253 (54740)

621 (53839)

3962 (49750)

758 (47832)

199  | 9  | 2, 5  |  | none - proven (for the k=3 prime, factor N+1 is equivalent to factor 199^183+1)  | 3 (183)

2 (16)

5 (6)

7 (3)

8 (2)

6 (2)

1 (2)

4 (1)

200  | 47  | 3, 13, 17  | k = 16:

odd n: factor of 3

n = = 0 mod 4: factor of 17

n = = 2 mod 4: let n = 4*q - 2 and let m = 20^q*10^(q-1); factors to:

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

| 1, 40 (k = 1 at n=16.77M, k = 40 at n=1M)  | 25 (21874)

10 (6036)

13 (1858)

38 (1669)

26 (1011)

5 (767)

34 (710)

19 (528)

46 (226)

43 (124)

256  | 38  | 3, 7, 13  | All k=4*q^4 for all n:

let k=4*q^4

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

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

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

| none - proven (primality certificate for k=11, primality certificate for k=23)  | 11 (5702)

23 (537)

20 (20)

7 (15)

22 (10)

25 (8)

15 (6)

36 (5)

6 (5)

28 (3)

| k = 4 proven composite by full algebraic factors.
512  | 18  | 5, 13, 19  | All k = m^3 for all n;

factors to:

(m*8^n + 1) *

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

| 2, 4, 5, 16 (k = 2 at n=2.001P, k = 4 at n=62.54T, k = 5 at n=1M, k = 16 at n=1.954T)  | 12 (23)

14 (21)

7 (20)

11 (9)

9 (7)

10 (6)

17 (3)

13 (2)

3 (2)

15 (1)

| k = 1 and 8 proven composite by full algebraic factors.
1024  | 81  | 5, 41  | All k = m^5 for all n;

factors to:

(m*4^n + 1) *

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

| 4, 16, 29, 38, 56 (k = 4 at n=858.9M, k = 16 at n=1.717G, other k at n=3K)  | 44 (1933)

41 (350)

9 (323)

51 (266)

14 (221)

33 (142)

48 (53)

11 (46)

54 (37)

10 (36)

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