generalized Fermat number

The numbers F_b,n = (with n and b integers, b greater than one) are called the generalized Fermat numbers because they are Fermat numbers in the special case b=2.

When b is even, these numbers share many properties with the regular Fermat numbers. For example, they have no algebraic factors; for a fixed base b they are all pairwise relatively prime; and all of the prime divisors have the form k^.2^m+1 with k odd and m > n. (When b is even, many of these properties are are shared by the numbers F_b,n/2.)

On the rare occasion that these generalized Fermat numbers are prime, they are call generalized Fermat primes.

See Also: Fermats, GeneralizedFermatPrime, Cullens, Mersennes

Related pages (outside of this work)

• The generalized Fermats on the list of 5000 largest known primes

References:

BR98

A. Björn and H. Riesel, "Factors of generalized Fermat numbers," Math. Comp., 67 (1998) 441--446.  MR 98e:11008 (Abstract available)

DG2000

H. Dubner and Y. Gallot, "Distribution of generalized Fermat prime numbers," Math. Comp., 71 (2002) 825--832.  MR 2002j:11156 (Abstract available)

DK95

H. Dubner and W. Keller, "Factors of generalized Fermat numbers," Math. Comp., 64 (1995) 397--405.  MR 95c:11010

Dubner86

H. Dubner, "Generalized Fermat primes," J. Recreational Math., 18 (1985-86) 279--280.  MR 2002j:11156

RB94

H. Riesel and A. Börn, Generalized Fermat numbers.  In "Mathematics of Computation 1943-1993: A Half-Century of Computational Mathematics," W. Gautschi editor, Proc. Symp. Appl. Math. Vol, 48, Amer. Math. Soc., Providence, RI, 1994.  pp. 583-587, MR 95j:11006