Generalized Fermat numbers a^(2^n) + 1 factorization (...) Generalized Fermat numbers a^(2^n) + b^(2^n)
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Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), pp. 441-446 (reference BR98 in the Prime Pages). Compiled by Wilfrid Keller. For an easy to read listing of a reduced segment of the above. tables supplement, see Jim Fougeron's additional pages. .
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Both definitions generalize the usual Fermat numbers F_n = 2 (2 n) +1. Generalized Fermat numbers can be prime only for even a. More specifically, an odd prime p is a generalized Fermat prime iff there exists an integer i with i 2 ≡ -1 (mod p) and i 2 < p (Broadhurst 2006).
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Welcome to the Generalized Fermat Prime Search! During the 17 th century, Pierre de Fermat and Marin Mersenne studied two particular forms of numbers, thinking that they could produce a large amount of prime numbers or even to be ever prime.
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Generalized Fermat numbers can be prime only for even a, because if a is odd then every generalized Fermat number will be divisible by 2. By analogy with the heuristic argument for the finite number of primes among the base-2 Fermat numbers, it is to be expected that there will be only finitely many generalized Fermat primes for each even base.
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A theorem of Édouard Lucas related to the Fermat numbers states that : Any prime divisor p of Fn = 22n + 1 is of the form p = k ⋅ 2n + 2 + 1 whenever n is greater than one. Does anyone know is there some similar theorem for generalized Fermat numbers: Fn(a)...
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Any generalized Fermat number F b,n = (with b an integer greater than one and n greater than zero) which is prime is called a generalized Fermat prime (because they are Fermat primes in the special case b=2).
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FACTORS OF GENERALIZED FERMAT NUMBERS 399. and this relation holds because n > m is assumed. We thus conclude that there are d = 2m solutions with b < P of (5) for each m . But (5) is the equivalent to saying that P divides Fb m for the base b in question (note that b = 0 and b = 1 can never occur).
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There are two different definitions of generalized Fermat numbers, one of which is more general than the other. Ribenboim (1996, pp. 89 and 359-360) defines a generalized Fermat number as a number of the form with , while Riesel (1994) further generalizes, defining it to be a number of the form .
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GIMPS has discovered a new largest known prime number: 2 82589933 -1 (24,862,048 digits) 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.
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A Fermat prime is a Fermat number F_n=2^(2^n)+1 that is prime. Fermat primes are therefore near-square primes. Fermat conjectured in 1650 that every Fermat number is prime and Eisenstein in 1844 proposed as a problem the proof that there are an infinite number of Fermat primes (Ribenboim 1996, p. 88).
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Factorization of generalized Fermat numbers for small indices m. Overview extending material contained in the paper Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), pp. 441-446 (reference BR98 in the Prime Pages). Compiled by Wilfrid Keller.
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