generalized Fermat prime
(another Prime Pages' Glossary entries)
The Prime Glossary
Glossary: Prime Pages: Top 5000:
GIMPS has discovered a new largest known prime number: 282589933-1 (24,862,048 digits)

Any prime generalized Fermat number Fb,n = b^2^n+1 (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 bn/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)


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

Chris K. Caldwell © 1999-2019 (all rights reserved)