Generalized Fermat Number (original) (raw)
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There are two different definitions of generalized Fermat numbers, one of which is more general than the other. Ribenboim (1996, pp. 89 and 359-360) defines a generalized Fermat number as a number of the form 
with 
, while Riesel (1994) further generalizes, defining it to be a number of the form 
. Both definitions generalize the usual Fermat numbers 
. The following table gives the first few generalized Fermat numbers for various bases
.
 |
OEIS | generalized Fermat numbers in base  |
|---|---|---|
| 2 | A000215 | 3, 5, 17, 257, 65537, 4294967297, ... |
| 3 | A059919 | 4, 10, 82, 6562, 43046722, ... |
| 4 | A000215 | 5, 17, 257, 65537, 4294967297, 18446744073709551617, ... |
| 5 | A078303 | 6, 26, 626, 390626, 152587890626, ... |
| 6 | A078304 | 7, 37, 1297, 1679617, 2821109907457, ... |
Generalized Fermat numbers can be prime only for even 
. More specifically, an odd prime 
is a generalized Fermat prime iff there exists an integer
with 
and 
(Broadhurst 2006).
Many of the largest known prime numbers are generalized Fermat numbers. Dubner found 
(
digits) and 
(
digits) in September 1992 (Ribenboim 1996, p. 360). The largest known as of January 2009 is 
(https://t5k.org/primes/page.php?id=84401), which has 
decimal digits.
The following table gives the first few generalized Fermat primes for various even bases 
.
See also
Fermat Number, Fermat Prime, Near-Square Prime
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References
Broadhurst, D. "GFN Conjecture." Post to primeform user forum. Apr. 1, 2006. http://groups.yahoo.com/group/primeform/message/7187Caldwell, C. "The Largest Known Primes." https://t5k.org/primes/lists/all.txt.Dubner, H. "Generalized Fermat Primes." J. Recr. Math. 18, 279-280, 1985.Dubner, H. and Keller, W. "Factors of Generalized Fermat Numbers."Math. Comput. 64, 397-405, 1995.Morimoto, M. "On Prime Numbers of Fermat Type." Sugaku 38, 350-354, 1986.Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, 1996.Riesel, H. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 102-103 and 415-428, 1994.Sloane, N. J. A. SequencesA000215/M2503, A059919,A078303, and A078304 in "The On-Line Encyclopedia of Integer Sequences."
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Cite this as:
Weisstein, Eric W. "Generalized Fermat Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GeneralizedFermatNumber.html