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

DK95

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