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Extended Sierpinski problems
Sierpinski problems
Definition 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 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 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 Colors used proven and all primes are defined primes
proven but some primes are only probable primes that have not been certified
unproven
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)
(only sorted by n)
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=31.8M)
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=175.6K, 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
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) *
(m2*4n - 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=33.55M, 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
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, 6476, 6553, 6598, 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
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) *
(m2*9n - 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=800K)
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, 43, 51, 73, 77, 107, 117, 149, 181, 209 (k = 1 at n=524K, other k at n=6K)
189 (5570)
191 (1553)
5 (1026)
113 (178)
121 (118)
145 (78)
37 (64)
33 (62)
205 (60)
97 (58)
32
10
3, 11
All k = m^5 for all n;
factors to:
(m*2^n + 1) *
(m4*16n - m3*8n + m2*4n - 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=87.8K)
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
344, 1098, 1169, 1229, 1415, 1600, 2012, 2215, 2294, 2338, 2543, 2768, 2789, 2951, 2957, 3050, 3281, 3656, 3689, 3812, 3935, 4127, 4224, 4388, 4468, 4514, 4565, 4586, 4675, 4742, 4757, 4820, 4835, 4883, 4943, 5003, 5042, 5126, 5165, 5372, 5414, 5477, 5698, 5700, 5944, 6014, 6095, 6376, 6413, 6563, 6689, 7051, 7076, 7092, 7172, 7299, 7319, 7404, 7552, 7586, 7707, 7934, 8117, 8165, 8255, 8273, 8283, 8324, 8362, 8363, 8552, 8624, 8792, 8978, 8980, 9090, 9101, 9221, 9224, 9238, 9731, 9935, 9964, 10112, 10187, 10261, 10639, 10652, 10661, 10690, 10741, 10762, 10988, 11112, 11192, 11195, 11293, 11306, 11356, 11358, 11438, 11522, 11635, 11645, 11684, 11750, 12164, 12422, 12668, 12791, 12955, 12994, 13025, 13094, 13193, 13283, 13324, 13406, 13445, 13904, 13970, 14103, 14465, 14510, 14555, 14679, 14730, 14759, 14816, 14909, 15104, 15130, 15263, 15284, 15292, 15374, 15417, 15579, 15581, 15702, 15803, 15989, 16235, 16319, 16445, 16481, 16768, 16850, 17303, 17465, 17477, 17957, 18083, 18146, 18164, 18285, 18365, 18386, 18398, 18410, 18491, 18572, 18613, 18692, 18695, 18779, 18818, 18859, 19037, 19073, 19187, 19202, 19213, 19280, 19394, 19570, 19640, 19884, 20051, 20124, 20198, 20213, 20214, 20267, 20318, 20376, 20402, 20540, 20870, 20894, 20951, 20963, 21026, 21032, 21176, 21196, 21207, 21407, 21895, 22016, 22057, 22136, 22327, 22426, 22467, 22671, 22945, 22961, 23042, 23123, 23189, 23201, 23246, 23342, 23371, 23479, 23492, 23582, 23621, 23741, 23799, 23816, 23984, 24085, 24167, 24221, 24437, 24476, 24519, 24594, 24599, 25337, 25501, 25624, 25667, 25799, 26006, 26036, 26075, 26198, 26241, 26255, 26387, 26731, 26815, 26855, 26921, 26947, 26987, 26990, 27102, 27182, 27389, 27430, 27464, 27614, 27653, 27948, 28332, 28382, 28496, 28535, 28552, 28578, 28619, 28778, 29045, 29108, 29150, 29291, 29342, 29603, 29642, 29849, 29972, 30227, 30236, 30269, 30344, 30503, 30505, 30546, 30608, 30647, 30751, 31079, 31088, 31220, 31226, 31418, 31489, 31538, 31733, 31770, 31928, 31952, 32078, 32206, 32375, 32512, 32555, 32637, 32660, 32678, 32717, 32756, 33065, 33158, 33170, 33211, 33344, 33482, 33581, 33662, 33764, 33785, 33827, 33913, 33929, 33959, 34029, 34175, 34505, 34646, 34709, 34748, 34808, 35188, 35333, 35375, 35382, 35384, 35390, 35417, 35429, 35507, 35519, 35546, 35552, 35612, 35669, 35822, 35828, 35835, 35837, 35894, 35999, 36011, 36101, 36163, 36170, 36185, 36243, 36368, 36436, 36655, 36668, 36808, 36824, 37205, 37229, 37268, 37358, 37391, 37514, 37577, 37703, 38023, 38047, 38084, 38252, 38306, 38324, 38334, 38378, 38664, 38763, 38825, 38828, 38900, 38951, 38980, 39014, 39115, 39119, 39180, 39230, 39525, 39722, 39743, 39853, 40438, 40517, 40667, 40878, 40940, 41165, 41411, 41444, 41450, 41479, 41695, 41696, 41750, 41798, 41819, 41999, 42106, 42230, 42473, 42899, 43019, 43058, 43174, 43295, 43334, 43499, 43727, 43787, 43830, 43892, 43994, 44238, 44279, 44447, 44546, 44617, 44665, 44732, 44759, 44894, 44969, 45222, 45272, 45676, 46337, 46370, 46698, 46709, 46862, 46925, 46987, 47155, 47272, 47276, 47429, 47559, 47561, 47582, 47684, 47693 (all at n=1K)
8870 (1000)
43254 (995)
44862 (981)
39533 (972)
40661 (967)
47069 (964)
8381 (963)
36983 (956)
2489 (946)
15118 (934)
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=600K)
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
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
42, 52, 106, 113, 158, 216, 266, 278, 311, 317, 366, 383, 419, 584, 608, 661, 674, 689, 743, 863, 902, 938, 941, 956, 973, 1043, 1100, 1241, 1247, 1292, 1324, 1326, 1376, 1378, 1433, 1463, 1483, 1502, 1538, 1591, 1642, 1658, 1689, 1727, 1730, 1778, 1808, 1907, 2150, 2174, 2297, 2378, 2384, 2386, 2396, 2516, 2570, 2598, 2624, 2632, 2711, 2813, 2894, 2978, 3107, 3114, 3181, 3232, 3254, 3386, 3418, 3426, 3434, 3474, 3497, 3602, 3659, 3671, 3746, 3749, 3767, 3827, 3868, 4007, 4073, 4112, 4133, 4135, 4241, 4292, 4373, 4706, 4804, 4901, 4928, 4967, 4970, 4981, 5087, 5281, 5282, 5343, 5354, 5399, 5405, 5567, 5570, 5573, 5619, 5621, 5624, 5633, 5693, 5711, 5723, 5725, 5776, 5831, 5882, 5909, 5912, 5988, 6002, 6011, 6037, 6044, 6125, 6147, 6149, 6239, 6246, 6331, 6359, 6385, 6536, 6572, 6632, 6654, 6687, 6743, 6767, 6770, 6836, 6891, 6981, 7058, 7089, 7147, 7207, 7237, 7262, 7283, 7313, 7358, 7397, 7400, 7577, 7580, 7586, 7653, 7737, 7739, 7763, 7883, 7990, 7998, 8048, 8054, 8132, 8189, 8255, 8322, 8331, 8392, 8479, 8579, 8638, 8681, 8693, 8723, 8786, 8948, 8973, 8983, 8990, 9083, 9134, 9150, 9242, 9243, 9314, 9329, 9356, 9380, 9421, 9433, 9437, 9542, 9563, 9602, 9635, 9698, 9737, 9848, 9943, 9977, 9988, 10002, 10004, 10013, 10061, 10154, 10172, 10188, 10192, 10246, 10328, 10396, 10411, 10451, 10487, 10493, 10499, 10548, 10586, 10601, 10641, 10652, 10667, 10679, 10739, 10793, 10853, 10861, 10862, 10916, 10917, 10919, 10946, 10971, 10999, 11042, 11078, 11120, 11138, 11146, 11237, 11321, 11391, 11516, 11522, 11553, 11684, 11714, 11747, 11765, 11771, 11798, 11818, 12035, 12062, 12091, 12191, 12197, 12201, 12266, 12391, 12404, 12461, 12471, 12533, 12623, 12721, 12779, 12884, 12918, 12931, 13043, 13088, 13136, 13152, 13171, 13251, 13277, 13310, 13316, 13355, 13362, 13451, 13478, 13491, 13514, 13673, 13697, 13728, 13784, 13799, 13808, 13842, 13922, 13952, 13994, 14129, 14132, 14234, 14256, 14336, 14447, 14583, 14657, 14691, 14786, 14849, 14888, 14906, 14998, 15110, 15123, 15157, 15282, 15422, 15424, 15474, 15545, 15617, 15636, 15637, 15656, 15659, 15687, 15737, 15901, 16046, 16058, 16119, 16133, 16166, 16204, 16219, 16273, 16352, 16442, 16481, 16535, 16559, 16571, 16574, 16607, 16652, 16661, 16738, 16742, 16749, 16802, 16853, 16893, 16961, 17012, 17022, 17027, 17054, 17120, 17165, 17167, 17168, 17247, 17277, 17279, 17342, 17383, 17491, 17543, 17573, 17712, 17723, 17809, 17819, 17996, 18072, 18077, 18233, 18236, 18251, 18328, 18449, 18458, 18526, 18602, 18604, 18632, 18636, 18686, 18724, 18797, 18816, 18857, 18914, 18951, 19043, 19066, 19081, 19094, 19121, 19132, 19157, 19178, 19241, 19319, 19328, 19337, 19352, 19397, 19403, 19451, 19493, 19556, 19592, 19634, 19646, 19721, 19751, 19768, 19872, 19959, 19967, 19980, 19982, 20035, 20163, 20192, 20300, 20351, 20459, 20475, 20487, 20516, 20526, 20624, 20722, 20830, 20840, 20897, 20936, 20975, 20987, 20996, 21041, 21136, 21167, 21212, 21246, 21272, 21347, 21353, 21354, 21359, 21517, 21653, 21701, 21806, 21835, 21851, 21902, 22024, 22053, 22055, 22071, 22169, 22233, 22332, 22418, 22430, 22457, 22479, 22526, 22685, 22701, 22709, 22719, 22727, 22739, 22787, 22791, 23007, 23062, 23222, 23374, 23531, 23558, 23586, 23612, 23641, 23659, 23663, 23705, 23743, 23774, 23805, 23844, 23871, 23886, 23902, 23906, 23929, 23947, 23984, 23987, 24169, 24257, 24273, 24328, 24347, 24374, 24448, 24452, 24456, 24464, 24497, 24547, 24563, 24697, 24708, 24722, 24866, 24911, 25070, 25123, 25176, 25227, 25229, 25236, 25439, 25471, 25492, 25494, 25558, 25616, 25619, 25653, 25704, 25757, 25847, 25865, 25874, 25876, 25932, 25943, 26009, 26067, 26072, 26078, 26128, 26210, 26222, 26261, 26287, 26300, 26322, 26498, 26513, 26548, 26614, 26658, 26660, 26744, 26771, 26813, 26858, 26923, 26966, 27031, 27082, 27122, 27296, 27327, 27479, 27516, 27519, 27527, 27572, 27623, 27642, 27718, 27720, 27743, 27764, 27779, 27837, 27877, 27879, 27983, 27985, 28079, 28142, 28193, 28198, 28208, 28211, 28229, 28277, 28333, 28462, 28493, 28658, 28661 (all at n=1K)
2474 (995)
20462 (992)
4285 (988)
10883 (985)
12968 (973)
15954 (962)
26722 (955)
4372 (954)
14444 (953)
13656 (953)
53
7
2, 3
4 (1.575M)
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
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, 133, 143, 185, 223, 237, 267, 283, 307, 309, 335, 343, 365, 367, 381, 391, 411, 425, 463, 467, 471, 487, 509, 549, 581, 587, 603, 605, 637, 643, 645, 673, 677, 681, 687, 689, 701, 789, 803, 807, 821, 825, 827, 881, 903, 937, 951, 963, 983, 989, 1021, 1027, 1043, 1047, 1049, 1063, 1067, 1103, 1121, 1189, 1201, 1207, 1263, 1267, 1283, 1321, 1341, 1367, 1401, 1433, 1461, 1463, 1467, 1481, 1523, 1553, 1563, 1581 (k = 1 at n=524K, other k at n=1K)
1108 (12351)
888 (2698)
9 (2162)
1174 (1989)
909 (938)
1085 (928)
1417 (918)
721 (816)
545 (810)
373 (774)
64
14
5, 13
All k = m^3 for all n;
factors to:
(m*4^n + 1) *
(m2*16n - m*4^n + 1)
none - proven
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
unknown
unknown
testing not started
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)
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
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)
79
9
2, 5
none - proven
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, 295, 326, 370, 393, 472, 556, 623, 628, 692, 778, 818, 947, 968 (k = 947 at n=4K, other k at n=250K)
188 (142291)
433 (121106)
770 (107149)
857 (106007)
787 (48156)
1024 (46306)
233 (36917)
893 (28705)
922 (21374)
683 (18633)
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)
34, 75, 239, 284, 317, 335, 389, 439, 514, 569 (all at n=1K)
558 (51992)
311 (7834)
41 (1223)
479 (495)
431 (414)
415 (385)
425 (258)
43 (236)
349 (227)
342 (218)
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
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
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)
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=1K)
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=300K)
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 (with probable primes that have not been certified: k = 191)
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, 198, 259, 412, 436, 449, 635, 653, 679, 740, 748, 812, 887, 929, 1000, 1088, 1160, 1190, 1421, 1429, 1511, 1544, 1559, 1607, 1628, 1703, 1796, 1823, 1835, 1925, 1973, 1985, 2018, 2036, 2075, 2119, 2177, 2189, 2216, 2279 (all at n=1K)
626 (998)
79 (987)
1001 (921)
632 (889)
1437 (807)
1310 (797)
890 (742)
1730 (720)
509 (695)
2330 (593)
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, 20543, 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, k = 20543 at n=2K, 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 kl 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
58, 59, 117 (k = 58 at n=250K, k = 59 at n=8K, k = 117 at n=524K)
75 (1428)
11 (1164)
77 (311)
2 (286)
81 (264)
47 (227)
67 (182)
4 (101)
51 (76)
109 (71)
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
unknown
unknown
testing not started
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=735K)
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
testing not started
125
7
2, 3
All k = m^3 for all n;
factors to:
(m*5^n + 1) *
(m2*25n - 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
testing not started
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) *
(m6*64n - m5*32n + m4*16n - m3*8n + m2*4n - 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.
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 (with probable primes that have not been certified: k = 11)
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) *
(m2*64n - m*8^n + 1)
2, 4, 5, 16 (k = 2 at n=7.635G, 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) *
(m4*256n - m3*64n + m2*16n - 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.