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Generalized Fermat Number


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 a^(2^n)+1 with a>2, while Riesel (1994) further generalizes, defining it to be a number of the form a^(2^n)+b^(2^n). Both definitions generalize the usual Fermat numbers F_n=2^(2^n)+1. The following table gives the first few generalized Fermat numbers for various bases a.

aOEISgeneralized Fermat numbers in base a
2A0002153, 5, 17, 257, 65537, 4294967297, ...
3A0599194, 10, 82, 6562, 43046722, ...
4A0002155, 17, 257, 65537, 4294967297, 18446744073709551617, ...
5A0783036, 26, 626, 390626, 152587890626, ...
6A0783047, 37, 1297, 1679617, 2821109907457, ...

Generalized Fermat numbers can be prime only for even a. More specifically, an odd prime p is a generalized Fermat prime iff there exists an integer i with i^2=-1 (mod p) and i^2<p (Broadhurst 2006).

Many of the largest known prime numbers are generalized Fermat numbers. Dubner found 200944^(2^(11))+1 (10861 digits) and 82642^(2^(11))+1 (10071 digits) in September 1992 (Ribenboim 1996, p. 360). The largest known as of January 2009 is 24518^(2^(18))+1 (http://primes.utm.edu/primes/page.php?id=84401), which has 1150678 decimal digits.

The following table gives the first few generalized Fermat primes for various even bases a.

aprime a^(2^n)+1
25, 17, 257, 65537, ...
417, 257, 65537, ...
637, 1297, ...

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/7187.Caldwell, C. "The Largest Known Primes." http://primes.utm.edu/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. Sequences A000215/M2503, A059919, A078303, and A078304 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Generalized Fermat Number

Cite this as:

Weisstein, Eric W. "Generalized Fermat Number." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GeneralizedFermatNumber.html

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