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  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://primes.utm.edu/primes/page.php?id=84401), which has  decimal digits.

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

	prime
2	5, 17, 257, 65537, ...
4	17, 257, 65537, ...
6	37, 1297, ...

REFERENCES:

Broadhurst, D. "GFN Conjecture." Post to primeform user forum. Apr. 1, 2006. https://groups.yahoo.com/group/primeform/message/7187.

Caldwell, C. "The Largest Known Primes." https://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."