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# generalized Fermat prime

Any prime generalized Fermat number F_{b,n}= (with

*b*an integer greater than one) is called a

**generalized Fermat prime**(because they are Fermat primes in the special case

*b*=2).

Why is the exponent a power of two?
Because if *m* is an odd divisor of *n*, then *b*^{n/m}+1 divides *b*^{n}+1, so for the latter to be
prime, *m* must be one. Because the exponent is a power of two, it seems reasonable to conjecture that
the number of Generalized Fermat primes is finite for
every fixed base *b*.

**See Also:** Fermats, Mersennes, Cullens

**Related pages** (outside of this work)

- Generalized Fermat Prime Search by Yves Gallot
- The top
twenty which lists the top 20 Generalized Fermat
divisor for several choices of
*b* - Smallest base values yielding Generalized Fermat Primes by Yves Gallot

**References:**

- BR98
A. BjörnandH. Riesel, "Factors of generalized Fermat numbers,"Math. Comp.,67(1998) 441--446.MR 98e:11008(Abstract available)- DK95
H. DubnerandW. Keller, "Factors of generalized Fermat numbers,"Math. Comp.,64(1995) 397--405.MR 95c:11010

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