Extended Riesel problems - xayahrainie4793/Extended-Sierpinski-Riesel-conjectures GitHub Wiki
Riesel problems
Definition For the original Riesel problem, it is finding and proving the smallest k such that k×bn-1 is not prime for all integers n ≥ 1 and GCD(k-1, b-1)=1.
Extended definiton 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 Riesel k
Covering set
k’s that make a full covering set with all or partial algebraic factors
Remaining k to find prime
(n testing limit)
Top 10 k’s with largest first primes: k(n)
(only sorted by n)
Comments
2
509203
3, 5, 7, 13, 17, 241
2293, 9221, 23669, 31859, 38473, 46663, 67117, 74699, 81041, 93839, 97139, 107347, 121889, 129007, 143047, 146561, 161669, 192971, 206039, 206231, 215443, 226153, 234343, 245561, 250027, 315929, 319511, 324011, 325123, 327671, 336839, 342847, 344759, 351134, 362609, 363343, 364903, 365159, 368411, 371893, 384539, 386801, 397027, 409753, 444637, 470173, 474491, 477583, 478214, 485557, 494743 (k = 351134 and 478214 at n=6.5M, other k at n=10M)
273809 (8932416)
502573 (7181987)
402539 (7173024)
40597 (6808509)
304207 (6643565)
398023 (6418059)
252191 (5497878)
353159 (4331116)
141941 (4299438)
123547 (3804809)
3
12119
2, 5, 7, 13, 73
1613, 1831, 1937, 3131, 3589, 5755, 6787, 7477, 7627, 7939, 8713, 8777, 9811, 10651, 11597 (all at n=50K)
8059 (47256)
11753 (36665)
6119 (28580)
7511 (26022)
313 (24761)
11251 (24314)
9179 (21404)
997 (20847)
6737 (17455)
7379 (16856)
4
361
3, 5, 7, 13
All k = m^2 for all n;
factors to:
(m*2^n - 1) *
(m*2^n + 1)
none - proven
106 (4553)
74 (1276)
219 (206)
191 (113)
312 (51)
247 (42)
223 (33)
274 (22)
234 (18)
91 (17)
k = 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, and 324 proven composite by full algebraic factors.
5
13
2, 3
none - proven
2 (4)
1 (3)
11 (2)
8 (2)
12 (1)
9 (1)
7 (1)
6 (1)
4 (1)
3 (1)
6
84687
7, 13, 31, 37, 97
1597, 6236, 9491, 37031, 49771, 50686, 53941, 55061, 57926, 76761, 79801, 83411 (k = 1597 at n=5.3M, other k at n=40K)
36772 (1723287)
43994 (569498)
77743 (560745)
51017 (528803)
57023 (483561)
78959 (458114)
59095 (171929)
48950 (143236),
29847 (141526)
9577 (121099)
7
457
2, 3, 5, 13, 19
none - proven (with probable primes that have not been certified: k = 197 and 367)
197 (181761)
367 (15118)
313 (5907)
159 (4896)
429 (3815)
419 (1052)
391 (938)
299 (600)
139 (468)
79 (424)
8
14
3, 5, 13
All k = m^3 for all n;
factors to:
(m*2^n - 1) *
(m2*4n + m*2^n + 1)
none - proven
11 (18)
5 (4)
12 (3)
7 (3)
2 (2)
13 (1)
10 (1)
9 (1)
6 (1)
4 (1)
k = 1 and 8 proven composite by full algebraic factors.
9
41
2, 5
All k = m^2 for all n;
factors to:
(m*3^n - 1) *
(m*3^n + 1)
none - proven
11 (11)
24 (8)
14 (8)
38 (3)
18 (3)
39 (2)
34 (2)
32 (2)
29 (2)
27 (2)
k = 1, 4, 9, 16, 25, and 36 proven composite by full algebraic factors.
10
334
3, 7, 13, 37
none - proven
121 (483)
109 (136)
98 (90)
230 (60)
289 (35)
89 (33)
32 (28)
233 (18)
324 (17)
100 (17)
11
5
2, 3
none - proven
1 (17)
3 (2)
2 (2)
4 (1)
12
376
5, 13, 29
(Condition 1):
All k where k = m^2
and m = = 5 or 8 mod 13:
for even n let k = m^2
and let n = 2*q; factors to:
(m*12^q - 1) *
(m*12^q + 1)
odd n:
factor of 13
(Condition 2):
All k where k = 3*m^2
and m = = 3 or 10 mod 13:
even n:
factor of 13
for odd n let k = 3*m^2
and let n=2*q-1; factors to:
[m*2(2q-1)*3q - 1] *
none - proven
298 (1676)
157 (285)
46 (194)
304 (40)
259 (40)
94 (36)
292 (30)
147 (28)
301 (27)
349 (25)
k = 25, 64, and 324 proven composite by condition 1.
k = 27 and 300 proven composite by condition 2.
13
29
2, 7
none - proven
25 (15)
28 (14)
20 (10)
1 (5)
22 (3)
17 (3)
16 (3)
27 (2)
21 (2)
12 (2)
14
4
3, 5
none - proven
2 (4)
1 (3)
3 (1)
15
622403
2, 17, 113, 1489
47, 203, 239, 407, 437, 451, 889, 893, 1945, 2049, 2245, 2487, 2507, 2689, 2699, 2863, 2940, 3059, 3163, 3179, 3261, 3409, 3697, 3701, 3725, 4173, 4249, 4609, 4771, 4877, 5041, 5243, 5425, 5441, 5503, 5669, 5857, 5913, 5963, 6231, 6447, 6787, 6879, 6999, 7386, 7407, 7459, 7473, 7527, 7615, 7683, 7687, 7859, 8099, 8610, 8621, 8671, 8839, 8863, 9025, 9267, 9409, 9655, 9663, 9707, 9817, 9955 (for k ⇐ 10K) (all at n=1.5K)
2940 (13254)
8610 (5178)
2069 (1461)
3917 (1427)
1145 (1349)
1583 (1330)
7027 (1316)
8831 (1296)
5305 (1273)
4865 (1265)
16
100
3, 7, 13
All k = m^2 for all n;
factors to:
(m*4^n - 1) *
(m*4^n + 1)
none - proven
74 (638)
78 (26)
48 (15)
58 (12)
31 (12)
95 (8)
46 (8)
88 (6)
44 (6)
39 (6)
k = 1, 4, 9, 16, 25, 36, 49, 64, and 81 proven composite by full algebraic factors.
17
49
2, 3
none - proven
44 (6488)
29 (4904)
13 (1123)
36 (243)
10 (117)
26 (110)
5 (60)
11 (46)
46 (25)
35 (24)
18
246
5, 13, 19
none - proven
151 (418)
78 (172)
50 (110)
79 (63)
237 (44)
184 (44)
75 (44)
215 (36)
203 (32)
93 (32)
19
9
2, 5
All k where k = m^2
and m = = 2 or 3 mod 5:
for even n let k = m^2
and let n = 2*q; factors to:
(m*19^q - 1) *
(m*19^q + 1)
odd n:
factor of 5
none - proven
1 (19)
7 (2)
3 (2)
8 (1)
6 (1)
5 (1)
2 (1)
k = 4 proven composite by partial algebraic factors.
20
8
3, 7
none - proven
2 (10)
1 (3)
6 (2)
5 (2)
7 (1)
4 (1)
3 (1)
21
45
2, 11
none - proven
29 (98)
34 (17)
43 (10)
32 (4)
5 (4)
6 (3)
1 (3)
44 (2)
37 (2)
31 (2)
22
2738
5, 23, 97
208, 211, 898, 976, 1036, 1885, 1933, 2050, 2161, 2278, 2347, 2434 (all at n=13K)
1013 (26067)
185 (11433)
1335 (11155)
2719 (9671)
2083 (8046)
883 (5339)
2529 (3700)
2116 (3371)
2230 (3236)
1119 (2849)
23
5
2, 3
none - proven
3 (6)
2 (6)
4 (5)
1 (5)
24
32336
5, 7, 13, 73, 577
(Condition 1):
All k where k = m^2
and m = = 2 or 3 mod 5:
for even n let k = m^2
and let n = 2*q; factors to:
(m*24^q - 1) *
(m*24^q + 1)
odd n:
factor of 5
(Condition 2):
All k where k = 6*m^2
and m = = 1 or 4 mod 5:
even n:
factor of 5
for odd n let k = 6*m^2
and let n=2*q-1; factors to:
[m*2(3q-1)*3q - 1] *
389, 461, 1581, 1711, 2094, 2606, 3006, 3754, 4239, 5356, 5784, 5791, 6116, 6579, 6781, 6831, 7321, 7809, 10219, 10399, 10666, 11101, 11516, 12326, 12429, 12674, 13269, 13691, 15019, 15151, 15614, 15641, 16124, 16234, 16616, 17019, 17436, 18054, 18454, 18964, 19116, 20026, 20576, 20611, 20879, 21004, 21464, 21524, 21639, 21809, 23549, 24404, 25046, 25136, 25349, 25389, 25419, 25646, 25731, 26176, 26229, 26661, 27049, 27154, 28001, 28384, 28849, 28859, 29211, 29531, 29569, 29581, 31071, 31466, 31734, 31854, 31994, 31996, 32099 (k = 1 mod 23 at n=12.4K, other k at n=260K)
10171 (259815)
11906 (252629)
23059 (252514)
21411 (252303)
28554 (239686)
20804 (233296)
8894 (210624)
2844 (203856)
25379 (175842)
22604 (169372)
k = 2^2, 3^2, 7^2, 8^2, 12^2, 13^2, 17^2, 18^2 (etc. pattern repeating every 5m) proven composite by condition 1.
k = 6*1^2, 6*4^2, 6*6^2, 6*9^2, 6*11^2, 6*14^2, 6*16^2, 6*19^2 (etc. pattern repeating every 5m) proven composite by condition 2.
25
105
2, 13
All k = m^2 for all n;
factors to:
(m*5^n - 1) *
(m*5^n + 1)
none - proven
86 (1029)
58 (26)
72 (24)
67 (24)
79 (21)
37 (17)
38 (14)
92 (13)
57 (10)
98 (9)
k = 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 proven composite by full algebraic factors.
26
149
3, 7, 31, 37
none - proven
115 (520277)
32 (9812)
121 (1509)
73 (537)
80 (382)
128 (300)
124 (249)
37 (233)
25 (133)
65 (100)
27
13
2, 7
All k = m^3 for all n;
factors to:
(m*3^n - 1) *
(m2*9n + m*3^n + 1)
none - proven
9 (23)
11 (10)
12 (2)
7 (2)
6 (2)
3 (2)
10 (1)
5 (1)
4 (1)
2 (1)
k = 1 and 8 proven composite by full algebraic factors.
28
3769
5, 29, 157
(Condition 1):
All k where k = m^2
and m = = 12 or 17 mod 29:
for even n let k = m^2
and let n = 2*q; factors to:
(m*28^q - 1) *
(m*28^q + 1)
odd n:
factor of 29
(Condition 2):
All k where k = 7*m^2
and m = = 5 or 24 mod 29:
even n:
factor of 29
for odd n let k = 7*m^2
and let n=2*q-1; factors to:
[m*2(2q-1)*7q - 1] *
233, 376, 943, 1132, 1422, 2437 (k = 233 and 1422 at n=1M, other k at n=20.3K)
2319 (65184)
3232 (9147)
3019 (7073)
460 (5400)
1688 (4760)
2406 (4634)
2464 (4324)
849 (3129)
1507 (2938)
472 (2414)
k = 144, 289, 1681, and 2116 proven composite by condition 1.
k = 175 proven composite by condition 2.
29
4
3, 5
none - proven
2 (136)
1 (5)
3 (1)
30
4928
13, 19, 31, 67
k = 1369:
for even n let n=2*q; factors to:
(37*30^q - 1) *
(37*30^q + 1)
odd n:
covering set 7, 13, 19
659, 1024, 1580, 1936, 2293, 2916, 3719, 4372, 4897 (all at n=500K)
1642 (346592)
239 (337990)
2538 (262614)
249 (199355)
3256 (160619)
225 (158755)
774 (148344)
1873 (50427)
3253 (43291)
1654 (38869)
31
145
2, 3, 7, 19
5, 19, 51, 73, 97 (all at n=6K)
123 (1872)
124 (1116)
113 (643)
49 (637)
115 (464)
21 (275)
39 (250)
70 (149)
142 (140)
33 (107)
32
10
3, 11
All k = m^5 for all n;
factors to:
(m*2^n - 1) *
(m4*16n + m3*8n + m2*4n + m*2^n + 1)
none - proven
3 (11)
2 (6)
9 (3)
8 (2)
5 (2)
7 (1)
6 (1)
4 (1)
k = 1 proven composite by full algebraic factors.
33
545
2, 17
(Condition 1):
All k where k = m^2
and m = = 4 or 13 mod 17:
for even n let k = m^2
and let n = 2*q; factors to:
(m*33^q - 1) *
(m*33^q + 1)
odd n:
factor of 17
(Condition 2):
All k where k = 33*m^2
and m = = 4 or 13 mod 17:
(Condition 3):
All k where k = m^2
and m = = 15 or 17 mod 32:
for even n let k = m^2
and let n = 2*q; factors to:
(m*33^q - 1) *
(m*33^q + 1)
odd n:
factor of 2
257, 339 (both at n=12K)
186 (16770)
254 (3112)
142 (2568)
370 (1628)
272 (1418)
222 (919)
108 (360)
213 (233)
387 (191)
277 (187)
k = 16, 169, and 441 proven composite by condition 1.
k = 528 proven composite by condition 2.
k = 225 and 289 proven composite by condition 3.
34
6
5, 7
All k where k = m^2
and m = = 2 or 3 mod 5:
for even n let k = m^2
and let n = 2*q; factors to:
(m*34^q - 1) *
(m*34^q + 1)
odd n:
factor of 5
none - proven
1 (13)
5 (2)
3 (1)
2 (1)
k = 4 proven composite by partial algebraic factors.
35
5
2, 3
none - proven
1 (313)
3 (6)
2 (6)
4 (1)
36
33791
13, 31, 43, 97
All k = m^2 for all n;
factors to:
(m*6^n - 1) *
(m*6^n + 1)
1148, 1555, 2110, 2133, 3699, 4551, 4737, 6236, 6883, 7253, 7362, 7399, 7991, 8250, 8361, 8363, 8472, 9491, 9582, 11014, 12320, 12653, 13641, 14358, 14540, 14836, 14973, 14974, 15228, 15687, 15756, 15909, 16168, 17354, 17502, 17946, 18203, 19035, 19646, 20092, 20186, 20630, 21880, 22164, 22312, 23213, 23901, 23906, 24236, 24382, 24645, 24731, 24887, 25011, 25159, 25161, 25204, 25679, 25788, 26160, 26355, 27161, 29453, 29847, 30970, 31005, 31634, 32302, 33047, 33627 (all at n=10K)
13800 (9790)
20485 (9140)
19389 (9119)
20684 (8627)
19907 (8439)
11216 (7524)
28416 (7315)
32380 (7190)
27296 (7115)
10695 (6672)
k = 1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2, 8^2, 9^2, 10^2, 11^2, 12^2, 13^2, 14^2, 15^2, 16^2, etc. proven composite by full algebraic factors.
37
29
2, 5, 7, 13, 67
none - proven
5 (900)
19 (63)
18 (14)
1 (13)
8 (4)
25 (3)
23 (3)
14 (3)
6 (3)
4 (3)
38
13
3, 5, 17
none - proven
11 (766)
9 (43)
7 (7)
1 (3)
12 (2)
8 (2)
5 (2)
2 (2)
10 (1)
6 (1)
39
9
2, 5
All k where k = m^2
and m = = 2 or 3 mod 5:
for even n let k = m^2
and let n = 2*q; factors to:
(m*39^q - 1) *
(m*39^q + 1)
odd n:
factor of 5
none - proven
1 (349)
7 (2)
3 (2)
2 (2)
8 (1)
6 (1)
5 (1)
k = 4 proven composite by partial algebraic factors.
40
25462
3, 7, 41, 223
(Condition 1):
All k where k = m^2
and m = = 9 or 32 mod 41:
for even n let k = m^2
and let n = 2*q; factors to:
(m*40^q - 1) *
(m*40^q + 1)
odd n:
factor of 41
(Condition 2):
All k where k = 10*m^2
and m = = 18 or 23 mod 41:
even n:
factor of 41
for odd n let k = 10*m^2
and let n=2*q-1; factors to:
[m*2(3q-1)*5q - 1] *
157, 490, 520, 534, 618, 709, 739, 787, 862, 955, 1067, 1114, 1174, 1242, 1352, 1544, 1559, 1762, 1795, 1805, 2254, 2290, 2830, 2887, 3033, 3034, 3156, 3342, 3361, 3418, 3650, 3750, 3830, 3859, 3922, 4006, 4132, 4183, 4219, 4297, 4582, 4673, 4724, 4771, 5218, 5233, 5308, 5431, 5629, 6107, 6192, 6220, 6436, 6463, 6582, 6618, 6682, 6684, 6709, 6946, 7089, 7094, 7126, 7258, 7282, 7381, 7504, 7602, 7678, 7702, 7795, 8032, 8035, 8173, 8461, 8572, 8899, 8959, 9121, 9226, 9347, 9424, 9472, 9511, 9716, 9748, 9874, 9964, 10003, 10060, 10285, 10615, 10744, 11030, 11470, 11479, 11560, 11847, 11971, 12178, 12193, 12226, 12250, 12256, 12299, 12301, 12422, 12505, 12544, 12547, 12568, 12709, 12742, 12750, 12873, 13005, 13022, 13039, 13165, 13191, 13212, 13624, 13666, 13777, 13894, 13939, 14146, 14262, 14272, 14362, 14494, 14513, 14636, 14766, 14802, 14980, 15046, 15154, 15271, 15374, 15376, 15388, 15417, 15496, 15579, 15661, 15730, 15907, 15967, 16108, 16235, 16579, 16705, 16728, 16891, 16897, 16932, 17014, 17137, 17275, 17287, 17344, 17536, 17653, 17707, 17801, 17860, 17896, 17923, 17998, 18114, 18292, 18397, 18697, 18787, 18818, 18853, 18949, 19117, 19310, 19510, 19606, 19722, 19751, 19756, 19761, 19780, 19825, 19927, 20158, 20212, 20253, 20428, 20458, 20479, 20491, 20583, 20632, 20747, 20788, 20809, 21058, 21082, 21276, 21321, 21403, 21493, 21731, 21817, 21895, 21975, 22114, 22130, 22262, 22263, 22303, 22344, 22570, 22706, 22879, 23371, 23615, 23851, 24184, 24189, 24268, 24337, 24397, 24421, 24448, 24483, 24519, 24805, 24979 (all at n=1K)
23977 (982)
13072 (982)
20952 (963)
8749 (962)
18103 (957)
22759 (939)
220 (939)
23795 (935)
8113 (918)
13654 (905)
k = 81, 1024, 2500, 5329, 8281, 12996, 17424, and 24025 proven composite by condition 1.
k = 3240 and 5290 proven composite by condition 2.
41
8
3, 7
none - proven
7 (153)
5 (10)
1 (3)
6 (2)
2 (2)
4 (1)
3 (1)
42
15137
5, 43, 353
603, 1049, 1600, 2538, 4299, 4903, 5118, 5978, 6836, 6964, 6971, 7309, 8297, 8341, 9029, 9201, 9633, 9848, 11267, 11781, 11911, 11996, 12125, 12127, 12213, 12598, 13288, 13347, 14884 (k = 1600, 6971 and 14884 at n=8K, other k at n=200K)
7051 (188034)
5417 (179220)
13898 (152983)
1633 (128734)
13757 (126934)
7913 (108747)
15024 (104613)
8453 (89184)
7658 (79316)
10923 (61071)
43
21
2, 11
13 (12K)
4 (279)
12 (203)
17 (79)
3 (24)
1 (5)
19 (4)
15 (4)
7 (4)
11 (2)
10 (2)
44
4
3, 5
none - proven
1 (5)
2 (4)
3 (1)
45
93
2, 23
none - proven
24 (153355)
53 (582)
70 (167)
29 (146)
76 (102)
85 (82)
91 (50)
77 (26)
1 (19)
33 (11)
46
928
3, 7, 103
281, 436, 800 (k = 800 at n=500K, other k at n=28K)
870 (51699)
86 (26325)
93 (24162)
561 (5011)
576 (3659)
100 (2955)
386 (2425)
338 (1478)
597 (950)
121 (935)
47
5
2, 3
none - proven
4 (1555)
1 (127)
2 (4)
3 (2)
48
3226
5, 7, 461
313, 384, 708, 909, 916, 1093, 1457, 1686, 1877, 1896, 1898, 2071, 2148, 2172, 2402, 2589, 2682, 2927, 2939, 3044, 3067 (all at n=200K)
2157 (169491)
2549 (169453)
1478 (167541)
2822 (129611)
2379 (116204)
118 (107422)
692 (103056)
1842 (87175)
953 (81493)
2582 (75696)
49
81
2, 5
All k = m^2 for all n;
factors to:
(m*7^n - 1) *
(m*7^n + 1)
none - proven
79 (212)
44 (122)
69 (42)
30 (24)
59 (16)
53 (15)
70 (14)
24 (14)
31 (9)
74 (6)
k = 1, 4, 9, 16, 25, 36, 49, and 64 proven composite by full algebraic factors.
50
16
3, 17
none - proven
14 (66)
13 (19)
5 (12)
11 (6)
6 (6)
1 (3)
8 (2)
2 (2)
15 (1)
12 (1)
51
25
2, 13
none - proven (with probable primes that have not been certified: k = 1)
1 (4229)
23 (96)
3 (8)
12 (4)
14 (3)
4 (3)
22 (2)
19 (2)
18 (2)
15 (2)
52
25015
3, 7, 53, 379
(Condition 1):
All k where k = m^2
and m = = 23 or 30 mod 53:
for even n let k = m^2
and let n = 2*q; factors to:
(m*52^q - 1) *
(m*52^q + 1)
odd n:
factor of 53
(Condition 2):
All k where k = 13*m^2
and m = = 7 or 46 mod 53:
even n:
factor of 53
for odd n let k = 13*m^2
and let n=2*q-1; factors to:
[m*2(2q-1)*13q - 1] *
82, 139, 233, 239, 349, 363, 372, 472, 476, 478, 547, 557, 607, 613, 654, 657, 796, 813, 902, 931, 991, 1012, 1069, 1104, 1161, 1167, 1231, 1234, 1271, 1357, 1502, 1534, 1589, 1591, 1651, 1669, 1711, 1801, 1881, 1909, 1966, 2035, 2049, 2113, 2227, 2364, 2384, 2437, 2492, 2557, 2578, 2643, 2722, 2725, 2767, 2769, 3022, 3073, 3106, 3128, 3163, 3199, 3229, 3277, 3298, 3418, 3423, 3550, 3559, 3607, 3637, 3656, 3764, 3788, 3847, 3870, 3921, 4003, 4036, 4043, 4117, 4195, 4239, 4294, 4329, 4347, 4348, 4366, 4534, 4561, 4582, 4597, 4665, 4754, 4762, 4824, 4876, 4894, 4975, 4981, 5037, 5056, 5107, 5142, 5158, 5236, 5239, 5246, 5299, 5541, 5575, 5672, 5836, 5882, 6190, 6193, 6256, 6308, 6361, 6394, 6424, 6434, 6442, 6462, 6493, 6568, 6589, 6619, 6628, 6697, 6732, 6835, 6873, 6962, 6981, 6997, 7252, 7288, 7386, 7399, 7408, 7594, 7603, 7631, 7633, 7727, 7797, 7799, 7847, 7879, 7894, 7936, 8008, 8032, 8138, 8161, 8163, 8201, 8248, 8257, 8377, 8389, 8422, 8488, 8587, 8637, 8641, 8691, 8693, 8713, 8744, 8903, 8932, 8958, 9053, 9055, 9144, 9148, 9187, 9223, 9382, 9400, 9421, 9433, 9436, 9472, 9624, 9647, 9654, 9667, 9682, 9699, 9753, 9769, 9782, 9799, 9802, 9808, 9854, 9859, 9892, 9907, 9928, 9967, 10056, 10069, 10129, 10134, 10173, 10174, 10237, 10243, 10306, 10429, 10462, 10489, 10546, 10618, 10645, 10792, 10806, 10917, 10919, 10954, 10984, 10996, 11161, 11164, 11290, 11297, 11299, 11326, 11355, 11371, 11394, 11401, 11500, 11656, 11677, 11698, 11722, 11767, 11826, 11827, 11833, 11854, 11926, 12064, 12074, 12133, 12148, 12186, 12212, 12239, 12304, 12352, 12401, 12405, 12423, 12449, 12454, 12668, 12688, 12694, 12719, 12827, 12889, 12928, 12931, 13025, 13031, 13045, 13196, 13198, 13264, 13297, 13306, 13324, 13357, 13372, 13392, 13461, 13551, 13673, 13687, 13719, 13786, 13856, 13999, 14044, 14065, 14101, 14116, 14179, 14234, 14266, 14309, 14453, 14584, 14589, 14647, 14682, 14692, 14698, 14736, 14759, 14786, 14827, 14833, 14947, 14968, 14998, 15007, 15010, 15022, 15051, 15109, 15124, 15139, 15154, 15181, 15212, 15244, 15265, 15316, 15370, 15574, 15677, 15688, 15733, 15899, 15928, 15937, 16007, 16039, 16087, 16096, 16111, 16216, 16227, 16293, 16308, 16324, 16342, 16388, 16429, 16535, 16614, 16714, 16726, 16729, 16748, 16836, 16854, 16884, 16897, 16906, 16927, 16963, 17092, 17102, 17182, 17197, 17224, 17229, 17277, 17311, 17418, 17423, 17438, 17489, 17714, 17734, 17754, 17782, 17821, 17882, 17911, 17916, 17989, 18604, 18670, 18709, 18757, 18761, 18787, 18871, 18883, 18899, 18903, 19024, 19026, 19028, 19079, 19098, 19102, 19132, 19142, 19163, 19189, 19282, 19357, 19363, 19549, 19556, 19558, 19594, 19609, 19672, 19678, 19821, 19876, 19946, 19982, 20008, 20088, 20094, 20139, 20212, 20267, 20308, 20318, 20359, 20395, 20417, 20616, 20649, 20793, 20821, 20881, 20883, 20983, 21013, 21016, 21049, 21092, 21148, 21151, 21235, 21307, 21316, 21368, 21403, 21404, 21413, 21464, 21526, 21537, 21572, 21676, 21684, 21729, 21757, 21784, 21796, 21803, 21804, 21837, 21859, 21866, 21898, 22096, 22146, 22180, 22216, 22308, 22312, 22324, 22383, 22406, 22429, 22447, 22456, 22459, 22471, 22528, 22566, 22643, 22688, 22704, 22723, 22738, 22744, 22771, 22789, 22842, 22846, 22874, 22887, 23056, 23191, 23215, 23268, 23315, 23344, 23377, 23427, 23518, 23531, 23533, 23584, 23692, 23759, 23773, 23829, 23924, 23991, 24042, 24175, 24244, 24331, 24403, 24412, 24448, 24503, 24553, 24557, 24591, 24646, 24671, 24763, 24911 (all at n=1K)
13298 (1000)
19006 (994)
10592 (993)
427 (992)
10687 (989)
14621 (982)
20044 (980)
8959 (980)
19084 (977)
k = 529, 900, 5776, 6889, 16641, and 18496 proven composite by condition 1.
k = 637 proven composite by condition 2.
53
13
2, 3
none - proven
12 (71)
10 (71)
2 (44)
7 (11)
1 (11)
8 (8)
11 (6)
9 (3)
5 (2)
6 (1)
54
21
5, 11
(Condition 1):
All k where k = m^2
and m = = 2 or 3 mod 5:
for even n let k = m^2
and let n = 2*q; factors to:
(m*54^q - 1) *
(m*54^q + 1)
odd n:
factor of 5
(Condition 2):
All k where k = 6*m^2
and m = = 1 or 4 mod 5:
even n:
factor of 5
for odd n let k = 6*m^2
and let n=2*q-1; factors to:
[m*2q*3(3q-1) - 1] *
none - proven
20 (8)
19 (6)
10 (4)
17 (3)
1 (3)
14 (2)
7 (2)
3 (2)
18 (1)
16 (1)
k = 4 and 9 proven composite by condition 1.
k = 6 proven composite by condition 2.
55
13
2, 7
none - proven
3 (76)
1 (17)
11 (8)
9 (3)
7 (2)
6 (2)
12 (1)
10 (1)
8 (1)
5 (1)
56
20
3, 19
none - proven
14 (26)
10 (23)
1 (7)
18 (4)
17 (4)
7 (3)
11 (2)
8 (2)
5 (2)
2 (2)
57
144
5, 13, 29
All k where k = m^2
and m = = 3 or 5 mod 8:
for even n let k = m^2
and let n = 2*q; factors to:
(m*57^q - 1) *
(m*57^q + 1)
odd n:
factor of 2
none - proven
87 (242)
54 (157)
100 (109)
59 (83)
115 (34)
124 (31)
88 (27)
63 (22)
139 (20)
38 (20)
k = 9, 25, and 121 proven composite by partial algebraic factors.
58
547
3, 7, 163
71, 130, 169, 178, 319, 456, 493, 499 (k = 71 and 456 at n=100K, other k at n=14K)
382 (7188)
400 (5245)
421 (4526)
176 (2854)
473 (1641)
487 (1412)
312 (1079)
334 (724)
53 (645)
457 (492)
59
4
3, 5
none - proven
3 (8)
1 (3)
2 (2)
60
20558
13, 61, 277
(Condition 1):
All k where k = m^2
and m = = 11 or 50 mod 61:
for even n let k = m^2
and let n = 2*q; factors to:
(m*60^q - 1) *
(m*60^q + 1)
odd n:
factor of 61
(Condition 2):
All k where k = 15*m^2
and m = = 22 or 39 mod 61:
even n:
factor of 61
for odd n let k = 15*m^2
and let n=2*q-1; factors to:
[m*2(2q-1)*15q - 1] *
36, 1770, 4708, 5317, 5611, 6101, 6162, 6274, 7060, 7870, 8722, 9212, 9454, 9881, 10249, 11101, 12061, 12072, 12098, 12479, 12996, 13297, 13480, 14275, 14851, 15800, 16167, 17185, 17620, 18055, 18965, 18972, 19336, 19394, 19397 (k = 16167 and 18055 at n=8K, other k at n=100K)
1024 (90701)
12121 (84208)
15227 (80625)
15185 (79350)
8649 (79159)
20131 (71977)
19457 (68854)
16333 (61172)
18776 (60164)
1486 (58932)
k = 121, 2500, 5184, 14641, and 17689 proven composite by condition 1.
k = 7260 proven composite by condition 2.
61
125
2, 31
37, 53, 100 (all at n=10K)
13 (4134)
77 (3080)
10 (1552)
41 (755)
42 (174)
22 (117)
57 (89)
109 (86)
103 (78)
93 (60)
62
8
3, 7
none - proven
3 (59)
4 (9)
1 (3)
6 (2)
5 (2)
2 (2)
7 (1)
63
857
2, 5, 397
37, 65, 93, 129, 139, 177, 211, 231, 237, 251, 271, 281, 291, 333, 417, 457, 471, 473, 491, 493, 497, 513, 587, 599, 633, 669, 677, 679, 687, 691, 695, 717, 733, 771, 817, 819, 821, 831, 853 (all at n=1K)
64 (1483)
372 (1320)
839 (940)
495 (916)
183 (904)
97 (851)
39 (848)
277 (835)
775 (710)
411 (678)
64
14
5, 13
All k = m^2 for all n; factors to:
(m*8^n - 1) *
(m*8^n + 1)
-or-
All k = m^3 for all n; factors to:
(m*4^n - 1) *
(m2*16n + m*4^n + 1)
none - proven
11 (9)
12 (6)
5 (2)
13 (1)
10 (1)
7 (1)
6 (1)
3 (1)
2 (1)
k = 1, 4, 8, and 9 proven composite by full algebraic factors.
65
10
3, 11
none - proven
1 (19)
8 (10)
4 (9)
2 (4)
5 (2)
9 (1)
7 (1)
6 (1)
3 (1)
66
unknown
unknown
testing not started
67
33
2, 17
All k where k = m^2
and m = = 4 or 13 mod 17:
for even n let k = m^2
and let n = 2*q; factors to:
(m*67^q - 1) *
(m*67^q + 1)
odd n:
factor of 17
none - proven
25 (2829)
2 (768)
23 (42)
21 (27)
1 (19)
31 (10)
19 (8)
18 (7)
13 (7)
11 (6)
k = 16 proven composite by partial algebraic factors.
68
22
3, 23
none - proven
7 (25395)
5 (13574)
11 (198)
8 (62)
10 (53)
3 (10)
1 (5)
14 (4)
2 (4)
9 (3)
69
6
3, 5
All k where k = m^2
and m = = 2 or 3 mod 5:
for even n let k = m^2
and let n = 2*q; factors to:
(m*69^q - 1) *
(m*69^q + 1)
odd n:
factor of 5
none - proven
5 (4)
1 (3)
3 (1)
2 (1)
k = 4 proven composite by partial algebraic factors.
70
853
13, 29, 71
376, 496, 811 (all at n=1K)
729 (28625)
434 (3820)
489 (2096)
278 (1320)
550 (764)
31 (545)
174 (441)
778 (356)
841 (335)
211 (329)
71
5
2, 3
none - proven
2 (52)
1 (3)
3 (2)
4 (1)
72
293
5, 17, 73
none - proven
4 (1119849)
79 (28009)
291 (26322)
116 (13887)
118 (4599)
67 (4308)
197 (3256)
24 (2648)
11 (2445)
18 (1494)
73
112
5, 13, 37
(Condition 1):
All k where k = m^2
and m = = 6 or 31 mod 37:
for even n let k = m^2
and let n = 2*q; factors to:
(m*73^q - 1) *
(m*73^q + 1)
odd n:
factor of 37
(Condition 2):
All k where k = m^2
and m = = 3 or 5 mod 8:
for even n let k = m^2
and let n = 2*q; factors to:
(m*73^q - 1) *
(m*73^q + 1)
odd n:
factor of 2
79, 101 (both at n=4K)
105 (102)
48 (73)
54 (63)
42 (50)
26 (50)
97 (47)
61 (39)
89 (32)
83 (26)
58 (25)
k = 36 proven composite by condition 1.
k = 9 and 25 proven composite by condition 2.
74
4
3, 5
none - proven
2 (132)
1 (5)
3 (2)
75
37
2, 19
none - proven
35 (1844)
16 (119)
18 (54)
30 (41)
3 (16)
22 (15)
5 (9)
17 (5)
4 (5)
23 (4)
76
34
7, 11
none - proven
1 (41)
27 (40)
20 (22)
25 (11)
15 (11)
30 (7)
21 (4)
19 (4)
13 (4)
10 (4)
77
13
2, 3
none - proven
2 (14)
1 (3)
12 (2)
11 (2)
8 (2)
5 (2)
3 (2)
10 (1)
9 (1)
7 (1)
78
90059
5, 79, 1217
274, 302, 631, 1816, 2292, 2381, 3872, 3949, 4344, 4383, 4489, 4937, 5057, 5766, 5782, 6077, 6436, 7032, 7800, 8469, 8499, 8649, 8758, 10263, 10924, 10928, 10942, 11044, 11936, 12167, 12187, 12244, 12286, 12332, 12622, 13212, 13287, 13668, 13824, 14059, 14456, 14526, 14932, 15722, 15799, 16451, 16688, 17029, 17039, 17221, 17271, 17732, 17886, 18013, 18663, 19614, 19846, 19909, 19986, 20027, 20182, 20462, 20879, 21197, 21631, 21961, 23052, 23079, 23801, 23899, 24214, 24949, 25061, 25532, 25901, 26377, 26385, 26804, 27021, 27096, 27175, 27256, 27399, 27439, 27842, 29073, 29389, 29668, 29863, 30444, 31046, 31053, 31742, 31836, 31917, 31994, 32705, 33298, 33412, 33671, 33888, 33892, 34728, 35179, 35568, 36233, 36344, 36609, 37024, 38354, 38438, 38711, 38886, 39173, 39901, 40131, 40239, 40289, 40437, 40998, 41079, 41316, 41711, 41748, 42106, 42337, 42896, 43331, 43842, 43886, 44038, 44374, 44634, 44871, 45214, 45221, 45466, 46012, 46187, 46593, 46922, 47004, 47562, 47573, 47636, 47657, 47986, 48004, 48112, 48371, 48973, 48979, 49386, 49611, 49988, 51430, 52042, 52929, 53719, 53761, 54188, 54936, 55245, 55491, 55617, 56563, 56721, 56757, 56904, 57234, 57317, 57611, 57786, 57842, 58402, 58455, 58696, 58854, 59093, 59536, 59774, 60187, 60919, 60978, 61762, 61783, 61937, 62481, 62646, 62854, 63043, 63281, 63351, 64309, 64384, 64744, 65157, 65814, 65885, 66102, 66249, 66991, 67386, 67588, 67593, 67706, 67880, 68027, 68573, 68804, 69630, 69914, 71254, 71338, 72003, 72916, 72997, 73706, 73708, 73734, 73787, 74757, 74823, 75307, 75482, 75857, 75888, 76056, 76392, 76781, 77057, 77594, 78135, 78604, 78835, 78959, 79630, 79633, 79674, 80421, 80725, 80788, 80976, 81208, 81369, 83186, 83739, 84484, 85218, 85506, 85886, 86137, 86164, 86329, 86353, 86446, 86692, 88718, 88817, 88866, 89314, 89538, 89664, 89846 (k = 1 mod 7 and k = 1 mod 11 at n=1K, other k at n=100K)
3633 (94500)
68571 (91386)
51476 (88677)
78053 (84433)
58412 (83824)
45661 (73022)
11412 (72798)
72638 (70230)
23462 (69162)
23543 (62677)
79
9
2, 5
All k where k = m^2
and m = = 2 or 3 mod 5:
for even n let k = m^2
and let n = 2*q; factors to:
(m*79^q - 1) *
(m*79^q + 1)
odd n:
factor of 5
none - proven
1 (5)
7 (4)
3 (4)
6 (3)
8 (1)
5 (1)
2 (1)
k = 4 proven composite by partial algebraic factors.
80
253
3, 37, 173
10, 31, 214 (all at n=400K)
170 (148256)
106 (16237)
154 (9753)
46 (5337)
232 (2997)
157 (2613)
169 (1959)
45 (1156)
218 (776)
244 (653)
81
74
7, 13, 73
All k = m^2 for all n;
factors to:
(m*9^n - 1) *
(m*9^n + 1)
none - proven
53 (268)
42 (99)
23 (68)
18 (15)
35 (14)
30 (12)
71 (4)
60 (4)
40 (4)
24 (4)
k = 1, 4, 9, 16, 25, 36, 49, and 64 proven composite by full algebraic factors.
82
22326
5, 83, 269
118, 133, 290, 331, 334, 439, 625, 649, 667, 748, 757, 763, 829, 878, 883, 898, 997, 1163, 1252, 1279, 1327, 1348, 1351, 1531, 1741, 1827, 1936, 1991, 2050, 2157, 2263, 2278, 2419, 2431, 2539, 2543, 2588, 2635, 2668, 2797, 2836, 2896, 2929, 2971, 2974, 3079, 3121, 3156, 3293, 3319, 3436, 3653, 3796, 3817, 4068, 4078, 4083, 4118, 4372, 4399, 4447, 4481, 4483, 4780, 4801, 4867, 4898, 4972, 5053, 5182, 5230, 5311, 5329, 5401, 5560, 5562, 5713, 5893, 5899, 5975, 6028, 6122, 6124, 6143, 6178, 6186, 6226, 6296, 6343, 6418, 6427, 6571, 6631, 6925, 6994, 7054, 7056, 7303, 7386, 7388, 7396, 7474, 7615, 7723, 7801, 7813, 7822, 7884, 7892, 7969, 8065, 8314, 8368, 8384, 8499, 8629, 8761, 8830, 8878, 8891, 8941, 9124, 9166, 9304, 9409, 9461, 9712, 9739, 9967, 9988, 10000, 10036, 10075, 10147, 10162, 10448, 10542, 10891, 10957, 11056, 11086, 11119, 11123, 11271, 11372, 11485, 11533, 11553, 11665, 11728, 11827, 11884, 11929, 12079, 12169, 12202, 12211, 12283, 12547, 12562, 12587, 12791, 13126, 13141, 13358, 13531, 13613, 13768, 13779, 13792, 13862, 13891, 14095, 14109, 14161, 14188, 14242, 14257, 14275, 14349, 14441, 14524, 14531, 14563, 14614, 14687, 14855, 14939, 14941, 14986, 15046, 15136, 15271, 15343, 15349, 15403, 15493, 15508, 15634, 15679, 15682, 15852, 15997, 16024, 16103, 16131, 16242, 16312, 16534, 16633, 16753, 16756, 16767, 16954, 17011, 17401, 17512, 17518, 17761, 17803, 17833, 17878, 18058, 18061, 18431, 18448, 18514, 18538, 18550, 18757, 19093, 19237, 19309, 19372, 19414, 19444, 19519, 19672, 19678, 19930, 19946, 20002, 20050, 20113, 20218, 20251, 20413, 20491, 20578, 20581, 20708, 20773, 20980, 21052, 21088, 21215, 21282, 21334, 21382, 21398, 21433, 21449, 21453, 21454, 21466, 21514, 21541, 21631, 21683, 21762, 21862, 21871, 21913, 22012, 22132, 22162, 22243, 22245 (k = 1 mod 3 at n=1K, other k at n=100K)
15978 (99999)
21429 (96772)
18989 (96049)
17592 (83837)
22233 (75716)
12912 (74869)
5811 (72615)
16091 (65850)
18576 (64927)
4482 (63245)
83
5
2, 3
none - proven
2 (8)
1 (5)
3 (2)
4 (1)
84
16
5, 17
All k where k = m^2
and m = = 2 or 3 mod 5:
for even n let k = m^2
and let n = 2*q; factors to:
(m*84^q - 1) *
(m*84^q + 1)
odd n:
factor of 5
none - proven
1 (17)
14 (8)
11 (7)
8 (4)
12 (3)
15 (1)
13 (1)
10 (1)
7 (1)
6 (1)
k = 4 and 9 proven composite by partial algebraic factors.
85
173
2, 43
61 (8K)
169 (6939)
64 (1253)
105 (403)
112 (394)
97 (287)
109 (230)
16 (171)
27 (160)
93 (90)
145 (77)
86
28
3, 29
none - proven
23 (112)
14 (38)
18 (26)
27 (14)
1 (11)
2 (10)
25 (9)
11 (8)
22 (5)
19 (5)
87
21
2, 11
none - proven
19 (372)
9 (91)
16 (17)
18 (15)
5 (15)
13 (11)
11 (10)
1 (7)
7 (6)
12 (5)
88
571
3, 7, 13, 19
k = 400:
for even n let n=2*q; factors to:
(20*88^q - 1) *
(20*88^q + 1)
odd n:
covering set 3, 7, 13
46, 49, 79, 94, 235, 277, 346, 508, 541, 544 (all at n=1K)
464 (20648)
444 (19708)
380 (8712)
477 (5816)
212 (5511)
179 (4545)
68 (2477)
536 (1731)
89 (1704)
17 (1362)
89
4
3, 5
none - proven
2 (60)
3 (5)
1 (3)
90
27
7, 13
All k where k = m^2
and m = = 5 or 8 mod 13:
for even n let k = m^2
and let n = 2*q; factors to:
(m*90^q - 1) *
(m*90^q + 1)
odd n:
factor of 13
none - proven
6 (20)
11 (10)
10 (10)
13 (6)
15 (5)
12 (4)
7 (4)
24 (3)
1 (3)
20 (2)
k = 25 proven composite by partial algebraic factors.
91
45
2, 23
none - proven (with probable primes that have not been certified: k = 1 and 27)
27 (5048)
1 (4421)
37 (159)
15 (14)
43 (6)
39 (6)
31 (6)
24 (5)
20 (4)
36 (3)
92
32
3, 31
none - proven
1 (439)
29 (272)
28 (99)
13 (35)
14 (32)
18 (26)
22 (25)
20 (6)
6 (6)
17 (4)
93
189
2, 47
33, 69, 109, 113, 125, 149, 177 (all at n=8K)
97 (1179)
29 (496)
92 (476)
46 (434)
121 (271)
141 (262)
101 (142)
122 (126)
85 (86)
166 (66)
94
39
5, 19
All k where k = m^2
and m = = 2 or 3 mod 5:
for even n let k = m^2
and let n = 2*q; factors to:
(m*94^q - 1) *
(m*94^q + 1)
odd n:
factor of 5
29 (1M)
16 (21951)
37 (254)
13 (163)
14 (154)
7 (95)
34 (54)
25 (41)
24 (12)
26 (9)
36 (7)
k = 4 and 9 proven composite by partial algebraic factors.
95
5
2, 3
none - proven
1 (7)
3 (2)
2 (2)
4 (1)
96
38995
7, 67, 97, 1303
(Condition 1):
All k where k = m^2
and m = = 22 or 75 mod 97:
for even n let k = m^2
and let n = 2*q; factors to:
(m*96^q - 1) *
(m*96^q + 1)
odd n:
factor of 97
(Condition 2):
All k where k = 6*m^2
and m = = 9 or 88 mod 97:
even n:
factor of 97
for odd n let k = 6*m^2
and let n=2*q-1; factors to:
[m*2(5q-1)*3q - 1] *
431, 486, 591, 701, 831, 872, 956, 1006, 1126, 1648, 1681, 1810, 2036, 2386, 2424, 2878, 3001, 3431, 3461, 3671, 3856, 3881, 3956, 3996, 4261, 4351, 4366, 4406, 4451, 4461, 5046, 5625, 5836, 5918, 6031, 6261, 6481, 6586, 6670, 6786, 7091, 7116, 7121, 7131, 7249, 7274, 7461, 7801, 8016, 8202, 8291, 8546, 8816, 9022, 9131, 9156, 9216, 9326, 9441, 9463, 9476, 9677, 9681, 9921, 10036, 10204, 10375, 10453, 10551, 10651, 10721, 11056, 11156, 11196, 11458, 11553, 11766, 11831, 12676, 12901, 13216, 13231, 13288, 13571, 14011, 14061, 14161, 14276, 14517, 14551, 14646, 15341, 15461, 15573, 15596, 16176, 16306, 16392, 16586, 16641, 16645, 17116, 17421, 17636, 17653, 17792, 18311, 19136, 19191, 19246, 19486, 19681, 20091, 20396, 20464, 20502, 20936, 21488, 21776, 22541, 22811, 22846, 22931, 23010, 23161, 23271, 23301, 23570, 23766, 24076, 24216, 24386, 24506, 24831, 24916, 24929, 25306, 25706, 25966, 26038, 26161, 26183, 26571, 26772, 26801, 26846, 27045, 27106, 27126, 27450, 27646, 27700, 27741, 28365, 28558, 28774, 28776, 28921, 29093, 29196, 29561, 29584, 29681, 30086, 30120, 30151, 30421, 30581, 30662, 31021, 31136, 31936, 32205, 32881, 33099, 33141, 33391, 33406, 33501, 33621, 33701, 33711, 33951, 33986, 34116, 34236, 34436, 34531, 34921, 35016, 35113, 35271, 35406, 35446, 35781, 35966, 36158, 36551, 36945, 36981, 37031, 37036, 37166, 37222, 37471, 37991, 38156, 38301, 38316, 38986 (k = 1 mod 5 and k = 1 mod 19 at n=1K, other k at n=100K)
3769 (92879)
28907 (89447)
13528 (86114)
19882 (82073)
37155 (76817)
9160 (71178)
5179 (66965)
32960 (60312)
7565 (59052)
4754 (56909)
k = 484, 5625, 14161, and 29584 proven composite by condition 1.
k = 486 proven composite by condition 2.
97
43
3, 5, 7, 37, 139
22 (35.8K)
8 (192335)
16 (1627)
4 (621)
28 (184)
1 (17)
34 (16)
32 (9)
27 (8)
37 (5)
31 (5)
98
10
3, 11
none - proven
1 (13)
5 (10)
7 (3)
4 (3)
8 (2)
2 (2)
9 (1)
6 (1)
3 (1)
99
9
2, 5
All k where k = m^2
and m = = 2 or 3 mod 5:
for even n let k = m^2
and let n = 2*q; factors to:
(m*99^q - 1) *
(m*99^q + 1)
odd n:
factor of 5
none - proven
5 (135)
3 (4)
1 (3)
7 (2)
8 (1)
6 (1)
2 (1)
k = 4 proven composite by partial algebraic factors.
100
211
7, 13, 37
All k = m^2 for all n;
factors to:
(m*10^n - 1) *
(m*10^n + 1)
none - proven (with probable primes that have not been certified: k = 133)
74 (44709)
133 (5496)
102 (209)
193 (155)
203 (133)
95 (96)
109 (68)
55 (56)
98 (45)
37 (36)
k = 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, and 196 proven composite by full algebraic factors.
101
13
2, 3
none - proven
5 (350)
8 (112)
2 (42)
11 (24)
12 (11)
4 (3)
1 (3)
6 (2)
10 (1)
9 (1)
102
1635
7, 19, 79
191, 207, 1082, 1369 (all at n=500K)
1451 (188973)
1208 (178632)
653 (117255)
1607 (82644)
254 (58908)
1527 (49462)
1037 (43460)
32 (43302)
1296 (37715)
142 (22025)
103
25
2, 13
none - proven
19 (820)
22 (442)
23 (216)
14 (189)
16 (57)
11 (54)
24 (32)
15 (32)
1 (19)
20 (5)
104
4
3, 5
none - proven
1 (97)
2 (68)
3 (1)
105
297
2, 37, 149
All k where k = m^2
and m = = 3 or 5 mod 8:
for even n let k = m^2
and let n = 2*q; factors to:
(m*57^q - 1) *
(m*57^q + 1)
odd n:
factor of 2
73, 137 (both at n=8K)
148 (3645)
265 (1666)
162 (294)
255 (222)
154 (139)
145 (119)
80 (91)
68 (56)
66 (47)
223 (21)
k = 9, 25, 121, and 169 proven composite by partial algebraic factors.
106
13624
3, 19, 199
64, 81, 163, 332, 391, 400, 429, 511, 526, 582, 596, 643, 676, 841, 862, 897, 913, 1024, 1223, 1261, 1283, 1294, 1417, 1428, 1546, 1597, 1713, 1869, 2056, 2116, 2248, 2389, 2458, 2605, 2623, 2656, 2674, 2719, 2743, 2780, 2781, 2813, 2888, 2965, 3047, 3073, 3130, 3136, 3142, 3241, 3277, 3336, 3425, 3427, 3478, 3481, 3617, 3622, 3646, 3655, 3694, 3746, 3883, 4045, 4067, 4096, 4153, 4162, 4177, 4219, 4336, 4339, 4416, 4628, 4662, 4666, 4696, 4713, 4722, 4801, 5135, 5283, 5359, 5395, 5468, 5485, 5623, 5692, 5707, 5752, 5776, 5777, 5872, 5878, 5937, 5971, 5992, 5993, 6040, 6094, 6100, 6103, 6181, 6220, 6376, 6421, 6505, 6547, 6613, 6716, 6736, 6832, 6955, 7069, 7156, 7202, 7246, 7273, 7297, 7331, 7336, 7345, 7356, 7398, 7402, 7496, 7540, 7561, 7744, 7771, 7894, 7906, 7915, 8023, 8181, 8266, 8323, 8329, 8371, 8386, 8428, 8521, 8561, 8572, 8637, 8779, 8788, 8861, 8950, 8956, 8962, 8975, 9031, 9096, 9190, 9238, 9294, 9366, 9415, 9469, 9589, 9634, 9736, 9774, 9787, 9790, 9796, 9808, 9859, 9877, 9973, 9976, 10033, 10072, 10117, 10150, 10166, 10186, 10271, 10273, 10446, 10451, 10627, 10646, 10651, 10660, 10699, 10816, 10876, 10894, 11097, 11173, 11278, 11299, 11419, 11420, 11426, 11506, 11639, 11671, 11833, 11884, 11901, 12066, 12076, 12090, 12145, 12252, 12269, 12321, 12352, 12361, 12490, 12627, 12851, 12856, 12910, 12916, 12970, 12978, 12991, 13023, 13027, 13162, 13174, 13269, 13366, 13374, 13378, 13387, 13497, 13511, 13516, 13528, 13543, 13553, 13558, 13567 (all at n=1K)
8272 (998)
508 (998)
13417 (994)
4908 (970)
5179 (969)
3700 (968)
577 (947)
3583 (943)
9814 (935)
1321 (913)
107
5
2, 3
none - proven (with probable primes that have not been certified: k = 3)
2 (21910)
3 (4900)
4 (251)
1 (17)
108
13406
7, 13, 61, 109
(Condition 1):
All k where k = m^2
and m = = 33 or 76 mod 109:
for even n let k = m^2
and let n = 2*q; factors to:
(m*108^q - 1) *
(m*108^q + 1)
odd n:
factor of 109
(Condition 2):
All k where k = 3*m^2
and m = = 20 or 89 mod 109:
even n:
factor of 109
for odd n let k = 3*m^2
and let n=2*q-1; factors to:
[m*2(2q-1)*3(3q-1) - 1] *
137, 411, 437, 873, 1634, 1769, 1782, 1961, 2508, 2617, 2962, 2963, 3002, 3029, 3474, 3499, 3596, 3646, 4007, 4066, 4084, 4121, 4184, 4328, 4468, 4499, 4744, 4904, 5015, 5142, 5212, 5351, 5625, 5821, 5892, 5923, 5994, 6212, 6284, 6432, 6528, 6570, 6614, 6866, 7107, 7211, 7302, 7304, 7419, 7848, 8037, 8144, 8374, 8383, 8503, 8524, 8638, 8986, 9346, 9852, 10052, 10129, 10136, 10245, 10699, 10926, 11089, 11164, 11278, 11619, 11881, 11918, 12262, 12861, 12863, 13162, 13291, 13297 (k = 5351, 6528, and 13162 at n=2K, other k at n=100K)
10322 (88080)
1999 (85188)
7557 (84180)
11882 (81547)
3439 (79524)
4686 (79010)
1159 (77107)
3573 (76352)
1465 (75209)
2148 (75018)
k = 1089 and 5776 proven composite by condition 1.
k = 1200 proven composite by condition 2.
109
9
2, 5
All k where k = m^2
and m = = 2 or 3 mod 5:
for even n let k = m^2
and let n = 2*q; factors to:
(m*109^q - 1) *
(m*109^q + 1)
odd n:
factor of 5
none - proven
8 (19)
1 (17)
5 (2)
2 (2)
7 (1)
6 (1)
3 (1)
k = 4 proven composite by partial algebraic factors.
110
38
3, 37
All k where k = m^2
and m = = 6 or 31 mod 37:
for even n let k = m^2
and let n = 2*q; factors to:
(m*110^q - 1) *
(m*110^q + 1)
odd n:
factor of 37
none - proven
23 (78120)
17 (2598)
37 (1689)
9 (77)
11 (42)
10 (17)
2 (16)
31 (9)
5 (6)
22 (5)
k = 36 proven composite by partial algebraic factors.
111
13
2, 7
none - proven
2 (24)
7 (6)
6 (4)
1 (3)
12 (2)
11 (2)
3 (2)
10 (1)
9 (1)
8 (1)
112
1357
5, 13, 113
All k where k = m^2
and m = = 15 or 98 mod 113:
for even n let k = m^2
and let n = 2*q; factors to:
(m*112^q - 1) *
(m*112^q + 1)
odd n:
factor of 113
31, 79, 310, 340, 421, 424, 451, 529, 703, 940, 1018, 1051, 1204 (all at n=7.5K)
948 (173968)
1268 (50536)
758 (35878)
1353 (7751)
187 (7524)
498 (6038)
9 (5717)
1024 (5681)
619 (5441)
981 (2858)
k = 225 proven composite by partial algebraic factors.
113
20
3, 19
none - proven
14 (308)
1 (23)
7 (15)
19 (11)
5 (8)
16 (5)
3 (5)
12 (3)
4 (3)
18 (2)
114
24
5, 23
All k where k = m^2
and m = = 2 or 3 mod 5:
for even n let k = m^2
and let n = 2*q; factors to:
(m*114^q - 1) *
(m*114^q + 1)
odd n:
factor of 5
none - proven
3 (63)
1 (29)
11 (27)
18 (21)
22 (20)
20 (3)
19 (2)
17 (2)
14 (2)
10 (2)
k = 4 and 9 proven composite by partial algebraic factors.
115
57
2, 29
13, 43 (both at n=8K)
45 (5227)
4 (4223)
51 (2736)
23 (1116)
53 (165)
21 (127)
35 (50)
15 (38)
39 (28)
32 (28)
116
14
3, 13
none - proven
9 (249)
5 (156)
11 (118)
1 (59)
2 (32)
13 (15)
10 (11)
12 (2)
8 (2)
7 (1)
117
149
2, 5, 37
5, 17, 33, 141 (all at n=8K)
83 (442)
59 (352)
19 (336)
110 (232)
143 (222)
41 (209)
87 (177)
129 (165)
118 (136)
92 (129)
118
50
7, 17
43 (37K)
27 (860)
29 (599)
18 (393)
6 (210)
22 (191)
8 (85)
19 (72)
7 (52)
42 (30)
37 (27)
119
4
3, 5
none - proven
2 (28)
3 (6)
1 (3)
120
unknown
unknown
testing not started
121
100
3, 7, 37
All k = m^2 for all n;
factors to:
(m*11^n - 1) *
(m*11^n + 1)
none - proven
62 (13101)
79 (4545)
43 (68)
7 (60)
30 (24)
60 (12)
87 (11)
39 (11)
57 (10)
50 (10)
k = 1, 4, 9, 16, 25, 36, 49, 64, and 81 proven composite by full algebraic factors.
122
14
3, 5, 13
none - proven
13 (43)
8 (26)
11 (10)
2 (6)
12 (5)
1 (5)
10 (3)
6 (2)
5 (2)
3 (2)
123
13
2, 5, 17
11 (8K)
1 (43)
3 (8)
2 (8)
12 (7)
6 (7)
9 (5)
7 (2)
10 (1)
8 (1)
5 (1)
124
92881
3, 5, 7, 5167
(Condition 1):
All k where k = m^2
and m = = 2 or 3 mod 5:
for even n let k = m^2
and let n = 2*q; factors to:
(m*124^q - 1) *
(m*124^q + 1)
odd n:
factor of 5
(Condition 2):
All k where k = 31*m^2
and m = = 1 or 4 mod 5:
even n:
factor of 5
for odd n let k = 31*m^2
and let n=2*q-1; factors to:
[m*2(2q-1)*31q - 1] *
testing not started
k = 2^2, 3^2, 7^2, 8^2, 12^2, 13^2, 17^2, 18^2 (etc. pattern repeating every 5m) proven composite by condition 1.
k = 31*1^2, 31*4^2, 31*6^2, 31*9^2, 31*11^2, 31*14^2, 31*16^2, 31*19^2 (etc. pattern repeating every 5m) proven composite by condition 2.
125
8
3, 7
All k = m^3 for all n;
factors to:
(m*5^n - 1) *
(m2*25n + m*5^n + 1)
none - proven
6 (24)
7 (5)
3 (3)
5 (2)
2 (2)
4 (1)
k = 1 proven composite by full algebraic factors.
126
480821
13, 19, 127, 829
testing not started
127
2593
2, 5, 17, 137
13, 17, 25, 27, 33, 35, 79, 83, 91, 113, 121, 139, 159, 179, 191, 231, 233, 235, 236, 237, 239, 250, 251, 264, 279, 288, 293, 333, 353, 361, 367, 379, 443, 451, 459, 471, 473, 511, 513, 517, 523, 531, 537, 551, 553, 557, 561, 597, 599, 604, 617, 631, 639, 649, 659, 679, 699, 715, 725, 731, 733, 737, 739, 747, 751, 755, 763, 773, 778, 783, 797, 809, 838, 848, 863, 871, 895, 919, 937, 939, 950, 953, 964, 982, 997, 999, 1013, 1019, 1025, 1031, 1037, 1039, 1043, 1051, 1106, 1107, 1117, 1119, 1127, 1157, 1173, 1185, 1196, 1199, 1211, 1231, 1232, 1233, 1245, 1253, 1259, 1279, 1288, 1291, 1313, 1327, 1333, 1335, 1337, 1347, 1353, 1359, 1371, 1377, 1401, 1407, 1417, 1421, 1429, 1432, 1439, 1473, 1481, 1491, 1513, 1525, 1539, 1549, 1551, 1573, 1577, 1579, 1589, 1593, 1595, 1597, 1599, 1611, 1612, 1618, 1631, 1639, 1641, 1661, 1677, 1693, 1699, 1709, 1711, 1731, 1732, 1737, 1751, 1771, 1792, 1793, 1803, 1837, 1839, 1903, 1911, 1921, 1928, 1933, 1936, 1939, 1943, 1951, 1957, 1959, 1999, 2013, 2017, 2032, 2039, 2045, 2072, 2073, 2079, 2092, 2097, 2099, 2129, 2155, 2168, 2179, 2191, 2197, 2215, 2231, 2247, 2253, 2273, 2279, 2303, 2313, 2339, 2367, 2377, 2389, 2411, 2427, 2431, 2433, 2479, 2501, 2543, 2548, 2559, 2565, 2573, 2583 (all at n=1K)
667 (1000)
1775 (994)
2497 (989)
2199 (972)
1759 (936)
2015 (910)
343 (904)
1113 (899)
1962 (893)
1543 (872)
128
44
3, 43
All k = m^7 for all n;
factors to:
(m*2^n - 1) *
(m6*64n + m5*32n + m4*16n + m3*8n + m2*4n + m*2^n + 1)
none - proven
29 (211192)
23 (2118)
26 (1442)
37 (699)
16 (459)
42 (246)
35 (98)
30 (66)
36 (59)
12 (46)
k = 1 proven composite by full algebraic factors.
256
100
3, 7, 13
All k = m^2 for all n;
factors to:
(m*16^n - 1) *
(m*16^n + 1)
none - proven
74 (319)
47 (228)
42 (224)
92 (143)
68 (87)
61 (54)
35 (28)
65 (24)
70 (18)
75 (17)
k = 1, 4, 9, 16, 25, 36, 49, 64, and 81 proven composite by full algebraic factors.
512
14
3, 5, 13
All k = m^3 for all n;
factors to:
(m*8^n - 1) *
(m2*64n + m*8^n + 1)
none - proven
4 (2215)
13 (2119)
9 (7)
11 (6)
6 (6)
5 (2)
3 (2)
2 (2)
12 (1)
10 (1)
k = 1 and 8 proven composite by full algebraic factors.
1024
81
5, 41
All k = m^2 for all n; factors to:
(m*32^n - 1) *
(m*32^n + 1)
-or-
All k = m^5 for all n;
factors to:
(m*4^n - 1) *
(m4*256n + m3*64n + m2*16n + m*4^n + 1)
29, 31, 56, 61 (k = 29 at n=1M, other k at n=3K)
74 (666084)
39 (4070)
43 (2290)
13 (1167)
78 (424)
65 (93)
69 (54)
3 (47)
71 (41)
44 (36)
k = 1, 4, 9, 16, 25, 32, 36, 49, and 64 proven composite by full algebraic factors.