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generalized Fermat number
The numbers Fb,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.2m+1 with k odd and m > n. (When b is even, many of these properties are are shared by the numbers Fb,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
- A. Björn and H. Riesel, "Factors of generalized Fermat numbers," Math. Comp., 67 (1998) 441--446. MR 98e:11008 (Abstract available)
- H. Dubner and Y. Gallot, "Distribution of generalized Fermat prime numbers," Math. Comp., 71 (2002) 825--832. MR 2002j:11156 (Abstract available)
- H. Dubner and W. Keller, "Factors of generalized Fermat numbers," Math. Comp., 64 (1995) 397--405. MR 95c:11010
- H. Dubner, "Generalized Fermat primes," J. Recreational Math., 18 (1985-86) 279--280. MR 2002j:11156
- 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., 1994. Providence, RI, pp. 583-587, MR 95j:11006
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