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# 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. BjörnandH. Riesel, "Factors of generalized Fermat numbers,"Math. Comp.,67(1998) 441--446.MR 98e:11008(Abstract available)- DG2000
H. DubnerandY. Gallot, "Distribution of generalized Fermat prime numbers,"Math. Comp.,71(2002) 825--832.MR 2002j:11156(Abstract available)- DK95
H. DubnerandW. 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. RieselandA. 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

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