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Status of the smallest base values yielding
Generalized Fermat primes
In 1986, Harvey Dubner published a table of the smallest integers b such
that GF(n, b) is prime (J. Recr. Math., 18,
1986). The following table reports Harvey Dubner's results and some
more recent results.
|
n |
N |
The first ten bases b for which GF(n, b) is prime |
| 1 |
2 |
2, [4], 6, 10, 14, [16], 20, 24, 26, [36], 40, 54, 56, ... |
| 2 |
4 |
2, [4], 6, [16], 20, 24, 28, 34, 46, 48, 54, 56, ... |
| 3 |
8 |
2, [4], 118, 132, 140, 152, 208, 240, 242, 288, 290, ... |
| 4 |
16 |
2, 44, 74, 76, 94, 156, 158, 176, 188, 198, ... |
| 5 |
32 |
30, 54, 96, 112, 114, 132, 156, 332, 342, 360, ... |
| 6 |
64 |
102, 162, 274, 300, 412, 562, 592, 728, 1084, 1094, ... |
| 7 |
128 |
120, 190, 234, 506, 532, 548, 960, 1738, 1786, 2884, ... |
| 8 |
256 |
278, 614, 892, 898, 1348, 1494, 1574, 1938, [2116], 2122, 2278, ... |
| 9 |
512 |
46, 1036, 1318, 1342, 2472, 2926, 3154, 3878, 4386, 4464, ... |
| 10 |
1024 |
824, 1476, 1632, 2462, 2484, 2520, 3064, 3402, 3820, 4026, ... |
| 11 |
2048 |
150, 2558, 4650, 4772, 11272, 13236, 15048, 23302, 26946, 29504, ... |
| 12 |
4096 |
1534, 7316, 17582, 18224, 28234, 34954, 41336, 48824, 51558, 51914, ... |
| 13 |
8192 |
30406, 71852, 85654, 111850, 126308, 134492, 144642, 147942, 150152, 165894,
... |
| 14 |
16384 |
67234, 101830, 114024, 133858, 162192, 165306, 210714, 216968, 229310, 232798,
... |
| 15 |
32768 |
70906, 167176, 204462, 249830, 321164, 330716,
332554, 429370, 499310, 524552, ... |
The test of all the GF(n, b) up to some large limits is
completed:
| n = 1, |
N = 2, |
B < 109,
|
34900212 Generalized Fermat primes
|
| n = 2, |
N = 4, |
B < 108,
|
3857543 Generalized Fermat primes |
| n = 3, |
N = 8, |
B < 107,
|
174368 Generalized Fermat primes |
| n = 4, |
N = 16, |
B < 107,
|
152447 Generalized Fermat primes |
| n = 5, |
N = 32, |
B < 107,
|
74951 Generalized Fermat primes |
| n = 6, |
N = 64, |
B < 107,
|
41059 Generalized Fermat primes |
| n = 7, |
N = 128, |
B < 9 106,
|
14586 Generalized Fermat primes |
| n = 8, |
N = 256, |
B < 7 106,
|
13890 Generalized Fermat primes |
| n = 9, |
N = 512, |
B < 6 106,
|
6138 Generalized Fermat primes |
| n = 10, |
N = 1024, |
B < 5 106,
|
2614 Generalized Fermat primes |
| n = 11, |
N = 2048, |
B < 4 106,
|
1000 Generalized Fermat primes |
| n = 12, |
N = 4096, |
B < 3 106,
|
435 Generalized Fermat primes |
| n = 13, |
N = 8192, |
B < 2.6 106,
|
>= 90 Generalized Fermat primes* |
| n = 14, |
N = 16384, |
B < 2.7 106,
|
>= 89 Generalized Fermat primes* |
| n = 15, |
N = 32768, |
B < 2.2 106,
|
>= 35 Generalized Fermat primes* |
| n = 16, |
N = 65536, |
B < 1.1 105,
|
>= 2 Generalized Fermat prime* |
*The range was tested only once. The number of primes will be
considered as correct when the range is double-checked. However, the
tests of significance indicate that the result is probably correct.
|
