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Generalized Fermat numbers
The generalized Fermat numbers (Riesel (1994)) are numbers of the form
The (less) generalized Fermat numbers (with ) (Ribenboim (1996)) are
where the ordinary Fermat numbers are
Contents
Generalized Fermat numbers a^(2^n) + 1
sequences | A-number | |
---|---|---|
2 | {3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, 115792089237316195423570985008687907853269984665640564039457584007913129639937, ...} | A000215 |
3 | {4, 10, 82, 6562, 43046722, 1853020188851842, 3433683820292512484657849089282, 11790184577738583171520872861412518665678211592275841109096962, ...} | A059919 |
4 | {5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, 115792089237316195423570985008687907853269984665640564039457584007913129639937, ...} | A000215 |
5 | {6, 26, 626, 390626, 152587890626, 23283064365386962890626, 542101086242752217003726400434970855712890626, ...} | A199591 |
6 | {7, 37, 1297, 1679617, 2821109907457, 7958661109946400884391937, 63340286662973277706162286946811886609896461828097, ...} | A078303 |
7 | {8, 50, 2402, 5764802, 33232930569602, 1104427674243920646305299202, 1219760487635835700138573862562971820755615294131238402, ...} | A078304 |
8 | {9, 65, 4097, 16777217, 281474976710657, 79228162514264337593543950337, 6277101735386680763835789423207666416102355444464034512897, ...} | A152581 |
9 | {10, 82, 6562, 43046722, 1853020188851842, 3433683820292512484657849089282, 11790184577738583171520872861412518665678211592275841109096962, ...} | A059919 |
10 | {11, 101, 10001, 100000001, 10000000000000001, 100000000000000000000000000000001, 10000000000000000000000000000000000000000000000000000000000000001, ...} | A080176 |
11 | {12, 122, 14642, 214358882, 45949729863572162, 2111377674535255285545615254209922, 4457915684525902395869512133369841539490161434991526715513934826242, ...} | A199592 |
12 | {13, 145, 20737, 429981697, 184884258895036417, 34182189187166852111368841966125057, 1168422057627266461843148138873451659428421700563161428957815831003137, ...} | A152585 |
13 | {14, 170, 28562, 815730722, 665416609183179842, ...} | A?????? |
14 | {15, 197, 38417, 1475789057, 2177953337809371137, 4743480741674980702700443299789930497, 22500609546641425009067997918450033531906583365663182830821882796510806017, ...} | A152587 |
15 | {16, 226, 50626, 2562890626, 6568408355712890626, ...} | A?????? |
16 | {17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, 115792089237316195423570985008687907853269984665640564039457584007913129639937, ...} | A000215 |
Formulae for a^(2^n) + 1
Generalized Fermat primes a^(2^n) + 1
Generalized Fermat numbers (Ribenboim (1996))
can't be prime if is odd.
Generalized Fermat primes
are congruent to 1 (mod 4).
Because of the ease of proving their primality, generalized Fermat primes have become in recent years a hot topic for research within the field of number theory. Many of the largest known primes[1] today are generalized Fermat primes.
By analogy with the heuristic argument for the finite number of primes among the base-2 Fermat numbers, it is to be expected that there will be only finitely many generalized Fermat primes for each even base. The smallest prime number with is .
A more elaborate theory can be used to predict the number of bases for which will be prime for a fixed . The number of generalized Fermat primes can be roughly expected to halve as is increased by 1.
sequences | A-number | |
---|---|---|
2 | {3, 5, 17, 257, 65537, ?} | A019434 |
4 | {5, 17, 257, 65537, ?} | A019434 |
6 | {7, 37, 1297, ?} | A?????? |
8 | (none) | A?????? |
10 | {11, 101, ?} | A?????? |
12 | {13, ?} | A?????? |
14 | {197, ?} | A?????? |
16 | {17, 257, 65537, ?} | A019434 |
18 | {19, ?} | A?????? |
20 | {401, ?} | A?????? |
22 | {23, ?} | A?????? |
24 | {577, ?} | A?????? |
Generalized Fermat numbers a^(2^n) + 1 factorization
(...)
Generalized Fermat numbers a^(2^n) + b^(2^n)
sequences | A-number | |||
---|---|---|---|---|
3 | 2 | 1 | {3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, 115792089237316195423570985008687907853269984665640564039457584007913129639937, ...} | A000215 |
4 | 3 | 1 | {4, 10, 82, 6562, 43046722, 1853020188851842, 3433683820292512484657849089282, 11790184577738583171520872861412518665678211592275841109096962, ...} | A059919 |
5 | 3 | 2 | {5, 13, 97, 6817, ...} | A?????? |
5 | 4 | 1 | {5, 17, 257, 65537, 4294967297, 18446744073709551617, ...} | A000215 |
6 | 4 | 2 | {6, 20, 272, 65792, ...} | A?????? |
6 | 5 | 1 | {6, 26, 626, 390626, 152587890626, 23283064365386962890626, 542101086242752217003726400434970855712890626, ...} | A199591 |
7 | 4 | 3 | {7, 25, 337, 72097, ...} | A?????? |
7 | 5 | 2 | {7, 29, 641, 390881, ...} | A?????? |
7 | 6 | 1 | {7, 37, 1297, 1679617, 2821109907457, 7958661109946400884391937, 63340286662973277706162286946811886609896461828097, ...} | A080174 |
8 | 5 | 3 | {8, 34, 706, 397186, ...} | A?????? |
8 | 6 | 2 | {8, 40, 1312, 1679872, ...} | A?????? |
8 | 7 | 1 | {8, 50, 2402, 5764802, 33232930569602, 1104427674243920646305299202, 1219760487635835700138573862562971820755615294131238402, ...} | A078304 |
9 | 5 | 4 | {9, 41, 881, 456161, ...} | A?????? |
9 | 6 | 3 | {9, 45, 1377, 1686177, ...} | A?????? |
9 | 7 | 2 | {9, 53, 2417, 5765057, ...} | A?????? |
9 | 8 | 1 | {9, 65, 4097, 16777217, 281474976710657, 79228162514264337593543950337, 6277101735386680763835789423207666416102355444464034512897, ...} | A152581 |
10 | 6 | 4 | {10, 52, 1552, 1745152, ...} | A?????? |
10 | 7 | 3 | {10, 58, 2482, 5771362, ...} | A?????? |
10 | 8 | 2 | {10, 68, 4112, 16777472, ...} | A?????? |
10 | 9 | 1 | {10, 82, 6562, 43046722, 1853020188851842, 3433683820292512484657849089282, 11790184577738583171520872861412518665678211592275841109096962, ...} | A059919 |
11 | 6 | 5 | {11, 61, 1921, 2070241, ...} | A?????? |
11 | 7 | 4 | {11, 65, 2657, 5830337, ...} | A?????? |
11 | 8 | 3 | {11, 73, 4177, 16783777, ...} | A?????? |
11 | 9 | 2 | {11, 85, 6577, 43046977, ...} | A?????? |
11 | 10 | 1 | {11, 101, 10001, 100000001, 10000000000000001, 100000000000000000000000000000001, 10000000000000000000000000000000000000000000000000000000000000001, ...} | A080176 |
12 | 7 | 5 | {12, 74, 3026, 6155426, ...} | A?????? |
12 | 8 | 4 | {12, 80, 4352, 16842752, ...} | A?????? |
12 | 9 | 3 | {12, 90, 6642, 43053282, ...} | A?????? |
12 | 10 | 2 | {12, 104, 10016, 100000256, ...} | A?????? |
12 | 11 | 1 | {12, 122, 14642, 214358882, 45949729863572162, 2111377674535255285545615254209922, 4457915684525902395869512133369841539490161434991526715513934826242, ...} | A199592 |
13 | 7 | 6 | {13, 85, 3697, 7444417, ...} | A?????? |
13 | 8 | 5 | {13, 89, 4721, 17167841, ...} | A?????? |
13 | 9 | 4 | {13, 97, 6817, 43112257, ...} | A?????? |
13 | 10 | 3 | {13, 109, 10081, 100006561, ...} | A?????? |
13 | 11 | 2 | {13, 125, 14657, 214359137, ...} | A?????? |
13 | 12 | 1 | {13, 145, 20737, 429981697, 184884258895036417, 34182189187166852111368841966125057, 1168422057627266461843148138873451659428421700563161428957815831003137, ...} | A152585 |
14 | 8 | 6 | {14, 100, 5392, 18456832, ...} | A?????? |
14 | 9 | 5 | {14, 106, 7186, 43437346, ...} | A?????? |
14 | 10 | 4 | {14, 116, 10256, 100065536, ...} | A?????? |
14 | 11 | 3 | {14, 130, 14722, 214365442, ...} | A?????? |
14 | 12 | 2 | {14, 148, 20752, 429981952, ...} | A?????? |
14 | 13 | 1 | {14, 170, 28562, 815730722, 665416609183179842, ...} | A?????? |
15 | 8 | 7 | {15, 113, 6497, 22542017, ...} | A?????? |
15 | 9 | 6 | {15, 117, 7857, 44726337, ...} | A?????? |
15 | 10 | 5 | {15, 125, 10625, 100390625, ...} | A?????? |
15 | 11 | 4 | {15, 137, 14897, 214424417, ...} | A?????? |
15 | 12 | 3 | {15, 153, 20817, 429988257, ...} | A?????? |
15 | 13 | 2 | {15, 173, 28577, 815730977, ...} | A?????? |
15 | 14 | 1 | {15, 197, 38417, 1475789057, 2177953337809371137, 4743480741674980702700443299789930497, 22500609546641425009067997918450033531906583365663182830821882796510806017, ...} | A152587 |
16 | 9 | 7 | {16, 130, 8962, 48811522, ...} | A?????? |
16 | 10 | 6 | {16, 136, 11296, 101679616, ...} | A?????? |
16 | 11 | 5 | {16, 146, 15266, 214749506, ...} | A?????? |
16 | 12 | 4 | {16, 160, 20992, 430047232, ...} | A?????? |
16 | 13 | 3 | {16, 178, 28642, 815737282, ...} | A?????? |
16 | 14 | 2 | {16, 200, 38432, 1475789312, ...} | A?????? |
16 | 15 | 1 | {16, 226, 50626, 2562890626, 6568408355712890626, ...} | A?????? |
17 | 9 | 8 | {17, 145, 10657, 59823937, ...} | A?????? |
17 | 10 | 7 | {17, 149, 12401, 105764801, ...} | A?????? |
17 | 11 | 6 | {17, 157, 15937, 216038497, ...} | A?????? |
17 | 12 | 5 | {17, 169, 21361, 430372321, ...} | A?????? |
17 | 13 | 4 | {17, 185, 28817, 815796257, ...} | A?????? |
17 | 14 | 3 | {17, 205, 38497, 1475795617, ...} | A?????? |
17 | 15 | 2 | {17, 229, 50641, 2562890881, ...} | A?????? |
17 | 16 | 1 | {17, 257, 65537, 4294967297, 18446744073709551617, ...} | A000215 |
Formulae for a^(2^n) + b^(2^n)
Generalized Fermat primes a^(2^n) + b^(2^n)
(...)
Generalized Fermat numbers a^(2^n) + b^(2^n) factorization
All factors of generalized Fermat numbers[2] (Riesel (1994))
are of the form
See also
Notes
- ↑ Chris K. Caldwell, The List of Largest Known Primes Home Page, (Another of the Prime Pages' resources) © 1999-2011.
- ↑ Anders Björn and Hans Riesel, Factors of Generalized Fermat Numbers, Mathematics of Computation, Vol. 67, No. 221, Jan., 1998, pp. 441-446.