Robinson Primes and the Sierpinski Problem
All primes must of necessity be of the form k.2n +1 for some odd k and some value of n. However, in view of the requirement for factors of Fermat Numbers, it is of interest to locate particular instances where either k is small and n is pushed as far as possible, or where n is small and k is pushed as far as possible. Viewed in this way, they are known as Robinson primes.
The following is a list of all values of n £ 4000 for each odd k < 100 such that k.2n + 1 is a prime. The number of such n for each k is also given.
|
k |
p |
n |
|
1 |
5 |
1, 2, 4, 8, 16 (Fermat Primes) |
|
3 |
24 |
1, 2, 5, 6, 8, 12, 18, 30, 36, 41, 66, 189, 201, 209, 276, 353, 408, 438, 534, 2208, 2816, 3168, 3189, 3912 |
|
5 |
13 |
1, 3, 7, 13, 15, 25, 39, 55, 75, 85, 127, 1947, 3313 |
|
7 |
21 |
2, 4, 6, 14, 20, 26, 50, 52, 92, 120, 174, 180, 190, 290, 320, 390, 432, 616, 830, 1804, 2256 |
|
9 |
31 |
1, 2, 3, 6, 7, 11, 14, 17, 33, 42, 43, 63, 65, 67, 81, 134, 162, 206, 211, 366, 663, 782, 1305, 1411, 1494, 2297, 2826, 3230, 3354, 3417, 3690 |
|
11 |
13 |
1, 3, 5, 7, 19, 21, 43, 81, 125, 127, 209, 211, 3225 |
|
13 |
10 |
2, 8, 10, 20, 28, 82, 188, 308, 316, 1000 |
|
15 |
26 |
1, 2, 4, 9, 10, 12, 27, 37, 38, 44, 48, 78, 112, 168, 229, 297, 339, 517, 522, 654, 900, 1518, 2808, 2875, 3128, 3888 |
|
17 |
12 |
3, 15, 27, 51, 147, 243, 267, 347, 471, 747, 2163, 3087 |
|
19 |
6 |
6, 10, 46, 366, 1246, 2038 |
|
21 |
23 |
1, 4, 5, 7, 9, 12, 16, 17, 41, 124, 128, 129, 187, 209, 276, 313, 397, 899, 1532, 1613, 1969, 2245, 2733 |
|
23 |
14 |
1, 9, 13, 29, 41, 49, 69, 73, 341, 381, 389, 649, 1961, 3929 |
|
25 |
24 |
2, 4, 6, 10, 20, 22, 52, 64, 78, 184, 232, 268, 340, 448, 554, 664, 740, 748, 1280, 1328, 1640, 3314, 3904, 3938 |
|
27 |
26 |
2, 4, 7, 16, 19, 20, 22, 26, 40, 44, 46, 47, 50, 56, 59, 64, 92, 175, 215, 275, 407, 455, 1076, 1090, 3080, 3322 |
|
29 |
20 |
1, 3, 5, 11, 27, 43, 57, 75, 77, 93, 103, 143, 185, 231, 245, 391, 1053, 1175, 2027, 3627 |
|
31 |
8 |
8, 60, 68, 140, 216, 416, 1808, 1944 |
|
33 |
21 |
1, 6, 13, 18, 21, 22, 25, 28, 66, 93, 118, 289, 412, 453, 525, 726, 828, 1420, 1630, 3076, 3118 |
|
35 |
20 |
1, 3, 7, 9, 13, 15, 31, 45, 47, 49, 55, 147, 245, 327, 355, 663, 1423, 1443, 2493, 3627 |
|
37 |
26 |
2, 4, 8, 10, 12, 16, 22, 26, 68, 82, 84, 106, 110, 166, 236, 254, 286, 290, 712, 1240, 1706, 1804, 1904, 2240, 2632, 3104 |
|
39 |
30 |
1, 2, 3, 5, 7, 10, 11, 13, 14, 18, 21, 22, 31, 42, 67, 70, 71, 73, 251, 370, 375, 389, 407,518, 818, 865, 1057, 1602, 2211, 3049 |
|
41 |
7 |
1, 11, 19, 215, 289, 379, 1991 |
|
43 |
17 |
2, 6, 12, 18, 26, 32, 94, 98, 104, 144, 158, 252, 778, 1076, 2974, 3022, 3528 |
|
45 |
16 |
2, 9, 12, 14, 23, 24, 29, 60, 189, 200, 333, 372, 443, 464, 801, 1374 |
|
47 |
2 |
583, 1483 |
|
49 |
12 |
2, 6, 10, 30, 42, 54, 66, 118, 390, 594, 1202, 2334 |
|
51 |
28 |
1, 3, 7, 9, 13, 17, 25, 43, 53, 83, 89, 92, 119, 175, 187, 257, 263, 267, 321, 333, 695, 825, 1485, 1917, 2660, 2967, 3447, 3659 |
|
53 |
14 |
1, 5, 17, 21, 61, 85, 93, 105, 133, 485, 857, 1665, 2133, 2765 |
|
55 |
15 |
4, 8, 16, 22, 32, 94, 220, 244, 262, 286, 344, 356, 392, 1996, 2744 |
|
57 |
22 |
2, 3, 7, 8, 10, 16, 18, 19, 40, 48, 55, 90, 96, 98, 190, 398, 456, 502, 719, 1312, 1399, 1828 |
|
59 |
7 |
5, 11, 27, 35, 291, 1085, 2685 |
|
61 |
6 |
4, 12, 48, 88, 168, 3328 |
|
63 |
37 |
1, 4, 5, 9, 10, 14, 17, 18, 21, 25, 37, 38, 44, 60, 65, 94, 133, 153, 228, 280, 314, 326, 334, 340, 410, 429, 626, 693, 741, 768, 1150, 1290, 1441, 2424, 2478, 3024, 3293 |
|
65 |
29 |
1, 3, 5, 11, 17, 21, 29, 47, 85, 93, 129, 151, 205, 239, 257, 271, 307, 351, 397, 479, 553, 1317, 1631, 1737, 1859, 1917, 1999, 2353, 3477 |
|
67 |
26 |
2, 6, 14, 20, 44, 66, 74, 102, 134, 214, 236, 238, 342, 354, 382, 454, 470, 598, 726, 870, 1148, 1366, 1692, 1782, 1870, 3602 |
|
69 |
17 |
1, 2, 10, 14, 19, 26, 50, 55, 145, 515, 842, 1450, 2159, 2290, 2306, 2335, 3379 |
|
71 |
14 |
3, 5, 9, 19, 23, 27, 57, 59, 65, 119, 299, 417, 705, 2255 |
|
73 |
16 |
2, 6, 14, 24, 30, 32, 42, 44, 60, 110, 212, 230, 1892, 1974, 2210, 3596 |
|
75 |
30 |
1, 3, 4, 6, 7, 10, 12, 34, 43, 51, 57, 60, 63, 67, 87, 102, 163, 222, 247, 312, 397, 430, 675, 831, 984, 1018, 1054, 1615, 2017, 2157 |
|
77 |
11 |
3, 7, 19, 23, 95, 287, 483, 559, 655, 667, 1639 |
|
79 |
9 |
2, 10, 46, 206, 538, 970, 1330, 1766, 2162 |
|
81 |
44 |
1, 4, 5, 7, 12, 15, 16, 21, 25, 27, 32, 35, 36, 39, 48, 89, 104, 121, 125, 148, 152, 267, 271, 277, 296, 324, 344, 396, 421, 436, 447, 539, 577, 592, 711, 809, 852, 1384, 1972, 2624, 2829, 3497, 3945, 3995 |
|
83 |
9 |
1, 5, 157, 181, 233, 373, 2425, 2773, 3253 |
|
85 |
18 |
4, 6, 10, 30, 34, 36, 38, 74, 88, 94, 148, 200, 624, 1300, 2458, 2556, 3638, 3834 |
|
87 |
17 |
2, 6, 8, 18, 26, 56, 78, 86, 104, 134, 207, 518, 602, 1268, 1302, 2324, 2372 |
|
89 |
11 |
1, 7, 9, 21, 37, 61, 589, 711, 1537, 1921, 3217 |
|
91 |
4 |
8, 168, 260, 696 |
|
93 |
27 |
2, 4, 6, 10, 12, 30, 42, 44, 52, 70, 76, 108, 122, 164, 170, 226, 298, 398, 686, 1020, 1110, 1478, 1646, 2032, 2066, 2800, 2816 |
|
95 |
25 |
1, 3, 5, 7, 13, 17, 21, 53, 57, 61, 83, 89, 111, 167, 175, 237, 533, 621, 661, 753, 993, 1039, 1849, 1987, 3437 |
|
97 |
12 |
2, 4, 14, 20, 40, 266, 400, 652, 722, 2026, 2732, 3880 |
|
99 |
33 |
1, 2, 5, 6, 10, 11, 22, 31, 33, 34, 41, 42, 53, 58, 65, 82, 126, 143, 162, 170, 186, 189, 206, 211, 270, 319, 369, 410, 433, 631, 894, 1617, 2025 |
The following is a list of the 50 odd k < 1000 for which there is no prime for n £ 10, and giving the lowest such n.
|
k |
n |
k |
n |
k |
n |
k |
n |
k |
n |
||||
|
47 |
583 |
259 |
38 |
383 |
6393 |
605 |
11 |
773 |
53 |
||||
|
103 |
16 |
263 |
29 |
389 |
11 |
607 |
12 |
797 |
35 |
||||
|
143 |
53 |
283 |
30 |
437 |
23 |
631 |
144 |
811 |
16 |
||||
|
197 |
15 |
335 |
19 |
451 |
12 |
647 |
15 |
827 |
19 |
||||
|
203 |
13 |
351 |
12 |
467 |
19 |
649 |
22 |
829 |
18 |
||||
|
211 |
20 |
353 |
21 |
481 |
64 |
689 |
15 |
881 |
1027 |
||||
|
217 |
66 |
361 |
28 |
533 |
13 |
733 |
12 |
901 |
12 |
||||
|
227 |
11 |
367 |
12 |
569 |
15 |
751 |
12 |
913 |
18 |
||||
|
241 |
36 |
377 |
11 |
587 |
227 |
767 |
23 |
917 |
27 |
||||
|
257 |
279 |
379 |
14 |
601 |
16 |
769 |
14 |
991 |
16 |
Table 1.
The following is a list of 104 odd k < 10000 for which there is no prime for n £ 100, giving the lowest such n, if found, and the search limit if not.
|
k |
n |
|
k |
n |
|
k |
n |
|
k |
n |
|
47 |
583 |
|
3331 |
172 |
|
5459 |
133 |
|
8021 |
119 |
|
257 |
279 |
|
3409 |
106 |
|
5483 |
137 |
|
8119 |
1162 |
|
383 |
6393 |
|
3443 |
3137 |
|
5771 |
167 |
|
8245 |
658 |
|
587 |
227 |
|
3529 |
122 |
|
5881 |
156 |
|
8269 |
1150 |
|
631 |
144 |
|
3533 |
261 |
|
5897 |
(20170) |
|
8423 |
(8000) |
|
881 |
1027 |
|
3589 |
118 |
|
6319 |
4606 |
|
8479 |
322 |
|
1021 |
112 |
|
3721 |
444 |
|
6353 |
785 |
|
8543 |
5793 |
|
1201 |
960 |
|
3797 |
315 |
|
6379 |
1014 |
|
8573 |
165 |
|
1277 |
143 |
|
3827 |
207 |
|
6439 |
578 |
|
8749 |
278 |
|
1523 |
145 |
|
3829 |
1230 |
|
6677 |
319 |
|
8791 |
268 |
|
1643 |
1465 |
|
3983 |
389 |
|
6719 |
103 |
|
8911 |
244 |
|
1669 |
230 |
|
4189 |
114 |
|
6721 |
208 |
|
8929 |
1966 |
|
1699 |
426 |
|
4283 |
689 |
|
6767 |
279 |
|
8933 |
261 |
|
1727 |
367 |
|
4337 |
103 |
|
6889 |
326 |
|
8957 |
115 |
|
1741 |
304 |
|
4367 |
715 |
|
6911 |
263 |
|
8989 |
418 |
|
1951 |
132 |
|
4429 |
342 |
|
7013 |
(24160) |
|
9049 |
262 |
|
2033 |
221 |
|
4727 |
227 |
|
7141 |
128 |
|
9053 |
241 |
|
2057 |
295 |
|
4801 |
752 |
|
7201 |
964 |
|
9091 |
204 |
|
2303 |
189 |
|
4847 |
(12062) |
|
7397 |
135 |
|
9181 |
136 |
|
2423 |
881 |
|
4861 |
2492 |
|
7489 |
290 |
|
9323 |
3013 |
|
2659 |
110 |
|
5207 |
251 |
|
7493 |
5249 |
|
9443 |
397 |
|
2831 |
105 |
|
5297 |
(12030) |
|
7651 |
(8000) |
|
9589 |
370 |
|
2897 |
9715 |
|
5359 |
(12069) |
|
7909 |
2174 |
|
9743 |
125 |
|
2987 |
123 |
|
5371 |
388 |
|
7913 |
125 |
|
9833 |
253 |
|
3061 |
(17007) |
|
5417 |
175 |
|
7921 |
156 |
|
9931 |
496 |
|
3289 |
274 |
|
5419 |
170 |
|
7957 |
5064 |
|
9953 |
205 |
Table 2.
n = 1 gives a prime for 24 out of the 50 values of odd k < 100.
n = 1 gives a prime for 155 out of the 500 values of odd k < 1000.
n = 1 gives a prime for 1136 out of the 5000 values of odd k < 10000.
n = 1 gives a prime for 9006 out of the 50000 values of odd k < 100000.
n = 2 gives a prime for 22 of the 50 values of odd k < 100.
n = 2 gives a prime for 140 of the 500 values of odd k < 1000.
n = 2 gives a prime for 1056 of the 5000 values of odd k < 10000.
n = 2 gives a prime for 8496 of the 50000 values of odd k < 100000.
n = 1 and 2 give a prime for 39 of the 50 values of odd k < 100.
n = 1 and 2 give a prime for 265 of the 500 values of odd k < 1000.
n = 1 and 2 give a prime for 2024 of the 5000 values of odd k < 10000.
n = 1 and 2 give a prime for 16484 of the 50000 values of odd k < 100000.
n = 1 to 10 give a prime for 49 of the 50 values of odd k < 100.
n = 1 to 10 give a prime for 450 of the 500 values of odd k < 1000.
n = 1 to 10 give a prime for 4150 of the 5000 values of odd k < 10000.
n = 1 to 10 give a prime for 38395 of the 50000 values of odd k < 100000.
n = 1 to 100 give a prime for 49 of the 50 values of odd k < 100.
n = 1 to 100 give a prime for 494 of the 500 values of odd k < 1000.
n = 1 to 100 give a prime for 4896 of the 5000 values of odd k < 10000.
n = 1 to 100 give a prime for 48706 of the 50000 values of odd k < 100000.
n = 1 to 1000 give a prime for 50 of the 50 values of odd k < 100.
n = 1 to 1000 give a prime for 498 of the 500 values of odd k < 1000.
n = 1 to 1000 give a prime for 4975 of the 5000 values of odd k < 10000.
n = 1 to 1000 give a prime for 49753 of the 50000 values of odd k < 100000.
A Sierpinski number is an odd number k such that k.2n +1 is composite for all n ³ 1. Sierpinski proved that the set of Sierpinski numbers is infinite, but the smallest examples he provided were of 18 digits. The Sierpinski Problem is the task of finding k0 - the smallest Sierpinski number. In his proof, Sierpinski used a covering set for each number, i.e. a finite set of primes such that every
k.2n +1 is divisible by at least one of the set. The smallest Sierpinski number known is k = 78557, with covering set {3,5,7,13,19,37,73}, though the Sierpinski number k = 271129 has the shortest covering set {3,5,7,13,17,241}, which has been proved to be the unique shortest possible. In attempting to find k0 , which may or may not be 78557, it is of particular interest to identify, for each odd k, values for which there are no primes for small n, and for each of these to keep raising the search limit on n until a prime is found.
The following is a list of 247 odd k < 100000 for which there is no prime for n £ 1000, giving the lowest value of n giving a prime, or the search limit. Note that the search uses trial division to remove as many values of n as possible from consideration, and so a search limit for a given prime is not necessarily a "round" number.
|
k |
n |
|
k |
n |
|
k |
n |
|
383 |
6393 |
|
10223 |
(8000) |
|
20851 |
(8000) |
|
881 |
1027 |
|
10583 |
2689 |
|
21143 |
1061 |
|
1643 |
1465 |
|
10967 |
2719 |
|
21167 |
6095 |
|
2897 |
9715 |
|
11027 |
1075 |
|
21181 |
(12091) |
|
3061 |
(17007) |
|
11479 |
1702 |
|
21901 |
1540 |
|
3443 |
3137 |
|
12395 |
1111 |
|
22699 |
(20133) |
|
3829 |
1230 |
|
12527 |
2435 |
|
22727 |
1371 |
|
4847 |
(12062) |
|
13007 |
1655 |
|
22951 |
1344 |
|
4861 |
2492 |
|
13787 |
(8000) |
|
23701 |
1780 |
|
5297 |
(12030) |
14027 |
(8000) |
|
23779 |
5234 |
|
|
5359 |
(12069) |
16519 |
3434 |
|
24151 |
2508 |
|
|
5897 |
(20170) |
16817 |
(8000) |
24737 |
(8000) |
||
|
6319 |
4606 |
16987 |
2748 |
24769 |
1514 |
||
|
6379 |
1014 |
17437 |
1812 |
24977 |
1079 |
||
|
7013 |
(24160) |
17597 |
3799 |
25171 |
2456 |
||
|
7493 |
5249 |
17629 |
1094 |
25339 |
4438 |
||
|
7651 |
(8000) |
17701 |
2700 |
25343 |
1989 |
||
|
7909 |
2174 |
18107 |
(12278) |
25819 |
(12001) |
||
|
7957 |
5064 |
18203 |
6141 |
25861 |
4848 |
||
|
8119 |
1162 |
19021 |
2608 |
26269 |
1086 |
||
|
8269 |
1150 |
19249 |
(18157) |
27653 |
(12344) |
||
|
8423 |
(8000) |
27923 |
(8000) |
||||
|
8543 |
5793 |
40547 |
(8000) |
28433 |
(12072) |
||
|
8929 |
1966 |
40553 |
1077 |
29629 |
1498 |
||
|
9323 |
3013 |
40571 |
1673 |
||||
|
41809 |
1402 |
50693 |
(8000) |
||||
|
30091 |
2184 |
42257 |
2667 |
51617 |
2675 |
||
|
31951 |
3084 |
42409 |
1506 |
51917 |
(8000) |
||
|
32161 |
(8000) |
43429 |
4290 |
52771 |
(8000) |
||
|
32393 |
4365 |
43471 |
1508 |
52909 |
3518 |
||
|
32731 |
1720 |
44131 |
(8000) |
53941 |
(8000) |
||
|
33661 |
(8000) |
44629 |
1270 |
54001 |
(12115) |
||
|
34037 |
1671 |
44903 |
(8000) |
54739 |
(8000) |
||
|
34565 |
3361 |
45713 |
1229 |
54767 |
(8000) |
||
|
34711 |
(8000) |
45737 |
2375 |
55459 |
(8000) |
||
|
34999 |
(12273) |
46157 |
(12046) |
56543 |
2501 |
||
|
35987 |
2795 |
46159 |
4790 |
56731 |
1172 |
||
|
36781 |
4824 |
46187 |
(8000) |
56867 |
1127 |
||
|
36983 |
(8000) |
46403 |
3057 |
57467 |
1259 |
||
|
37561 |
(8000) |
46471 |
(8000) |
57503 |
5697 |
||
|
38029 |
2778 |
47179 |
2918 |
57949 |
1058 |
||
|
39079 |
(12249) |
47897 |
(8000) |
58243 |
1136 |
||
|
39241 |
1120 |
47911 |
5568 |
59569 |
(8000) |
||
|
39781 |
(8000) |
48091 |
1476 |
||||
|
48323 |
1369 |
80419 |
1166 |
||||
|
60443 |
(12260) |
48833 |
(8000) |
80839 |
(4397) |
||
|
60541 |
(8000) |
49219 |
(8000) |
81091 |
(4631) |
||
|
60737 |
1411 |
81269 |
(4342) |
||||
|
60829 |
6398 |
70261 |
3048 |
81857 |
1399 |
||
|
61519 |
1290 |
71417 |
(8000) |
81871 |
1420 |
||
|
62093 |
(12016) |
71671 |
(8000) |
81919 |
(4817) |
||
|
62761 |
(8000) |
71869 |
5130 |
82267 |
2904 |
||
|
63017 |
(8000) |
72197 |
2171 |
82283 |
2469 |
||
|
63379 |
2070 |
73189 |
4278 |
82549 |
(4349) |
||
|
64007 |
(8000) |
73253 |
6889 |
82841 |
1037 |
||
|
64039 |
2246 |
73849 |
1202 |
82907 |
1475 |
||
|
65057 |
(8000) |
74191 |
(8000) |
84409 |
(4349) |
||
|
65477 |
5887 |
74221 |
4188 |
84887 |
1087 |
||
|
65539 |
1822 |
74269 |
(8000) |
85013 |
(4352) |
||
|
65567 |
(20154) |
74959 |
4274 |
85711 |
(4343) |
||
|
65623 |
1746 |
75841 |
(12211) |
86701 |
(4375) |
||
|
65791 |
2760 |
76261 |
2156 |
86747 |
(4346) |
||
|
65971 |
1224 |
76759 |
(8000) |
86761 |
3896 |
||
|
67193 |
(8000) |
76969 |
3702 |
86869 |
(4521) |
||
|
67607 |
(18170) |
77267 |
4159 |
87469 |
2110 |
||
|
67759 |
(8000) |
77341 |
5076 |
87743 |
(4352) |
||
|
67831 |
1720 |
77521 |
3336 |
88007 |
2843 |
||
|
67913 |
5773 |
77899 |
(12209) |
88157 |
1063 |
||
|
68393 |
1901 |
78181 |
(8000) |
88549 |
(4401) |
||
|
69107 |
(8000) |
78557 |
------- |
89003 |
1285 |
||
|
69109 |
(12021) |
79309 |
(5221) |
89059 |
(4433) |
||
|
79817 |
(4430) |
89225 |
(4384) |
||||
|
79819 |
1678 |
89521 |
1564 |
||||
|
79879 |
1098 |
89537 |
1203 |
||||
|
90019 |
3470 |
93331 |
(4463) |
97621 |
1820 |
||
|
90047 |
1403 |
93443 |
1277 |
97697 |
2139 |
||
|
90527 |
(4390) |
93593 |
3597 |
97789 |
1102 |
||
|
90529 |
2518 |
93617 |
(4570) |
98327 |
2911 |
||
|
90949 |
2254 |
94069 |
(4429) |
98431 |
(4327) |
||
|
91291 |
3432 |
94373 |
(4420) |
98461 |
1892 |
||
|
91399 |
1746 |
94639 |
2654 |
98723 |
1681 |
||
|
91549 |
(4445) |
95791 |
1088 |
98749 |
(4329) |
||
|
92119 |
(4521) |
96409 |
1426 |
99557 |
2931 |
||
|
92567 |
(4366) |
96983 |
(4344) |
99587 |
(4394) |
||
|
92749 |
1138 |
97555 |
(4355) |
99739 |
(4349) |
||
|
99769 |
1042 |
Table 3.
Counts for surviving k values in steps of 10000 are:-
|
|
|
total |
|
n £ 1000 : |
25, 21, 24, 18, 25, 18, 26, 26, 30, 34 |
247 |
|
n £ 2000 : |
18, 15, 14, 15, 16, 13, 19, 23, 20, 23 |
176 |
|
n £ 3000 : |
16, 9, 12, 12, 13, 11, 16, 21, 16, 17 |
143 |
|
n £ 4000 : |
14, 7, 12, 10, 12, 10, 16, 18, 15, 14 |
128 |
With direct reference to the Sierpinski Problem, only 178 odd k < 78557 survive with no primes of the form k.2n + 1 for n £ 1000, and this drops to 95 for n £ 4000. With further searches concentrated on these values, currently only 69 odd k < 78557 remain unaccounted for, the smallest being 3061. Therefore we can say that 3061 £ k0 £ 78557.
For each of the values of k > 78557 given in Table 3, only the lowest value of n giving a prime, if known, is listed. Since the search was pushed at least to n £ 4000 for each of these, the following large primes were discovered with n > 2000, of which 7 have more than 1000 digits.
|
k |
n |
|
79819 |
3598 |
|
81871 |
2164, 2956 |
|
82267 |
2904 |
|
82283 |
2469 |
|
86761 |
3896 |
|
87469 |
2110 |
|
88007 |
2843, 2915 |
|
90019 |
3470 |
|
90529 |
2518 |
|
90949 |
2254 |
|
91291 |
3432, 3672 |
|
93443 |
3633 |
|
93593 |
3597 |
|
94639 |
2654 |
|
97697 |
2139 |
|
98327 |
2911 |
|
98723 |
2245 |
|
99557 |
2931 |
There remain 31odd k in the higher range for which an associated Robinson prime is not known.
As suggested by the above, I have concentrated mainly in the range 78558 < k < 100000. As part of this, I have located all primes of the form k.2n + 1 for 1001 £ n £ 2000. I then extended this search to cover the range 70000 < k < 78556. Both ranges are detailed below.
range 1 : 78558 < k < 100000
range 2 : 70000 < k < 78556
|
range of n |
range 1 count |
range 2 count |
|
1001 - 1100 |
2856 |
1153 |
|
1101 - 1200 |
2715 |
1066 |
|
1201 - 1300 |
2435 |
972 |
|
1301 - 1400 |
2204 |
894 |
|
1401 - 1500 |
2144 |
839 |
|
1501 - 1600 |
1953 |
803 |
|
1601 - 1700 |
1875 |
744 |
|
1701 - 1800 |
1706 |
677 |
|
1801 - 1900 |
1610 |
626 |
|
1901 - 2000 |
1596 |
622 |
|
totals |
21094 |
8396 |
This gives at total of 29490 primes in the combined range.
The following three Fermat divisors that fall within this range were rediscovered and verified.
82165 * 21084 + 1 divides F1082 ; 79707 * 21231 + 1 divides F1225 ; 98855 * 21851 + 1 divides F1849
The highest number of primes for a particular k in the combined search range 70000 < k < 100000, and for 1001 £ n £ 2000, is 12. This occurs at k = 70365. A count of 11 is achieved for the k values 70653, 73413, 84435, 89595 and 91575.