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Generalized Fermat number
There are different kinds of generalized Fermat numbers.
Contents
John Cosgrave
John Cosgrave has studied the following numbers:
Numbers of the form: [math]\displaystyle{ F_{n,r} = \sum_{i=0}^{p-1} \ 2^{i p^{n}} \ = \ 2^{(p-1)p^n}+2^{(p-2)p^n}+...+2^{2p^n}+2^{p^n}+1 \ = \ (2^{p^{n+1}}-1)/(2^{p^n}-1) }[/math] where p is the prime of apparition rank r (r(2)=1, r(3)=2, r(5)=3, ...) and n is greater or equal to 0.
- [math]\displaystyle{ F_{0,r} }[/math] generates the Mersenne numbers.
- [math]\displaystyle{ F_{n,1} }[/math] generates the Fermat numbers.
- [math]\displaystyle{ F_{n,2} }[/math] generates the Saouter numbers.
Cosgrave has proven the following properties:
- If number [math]\displaystyle{ \sum_{i=0}^{p-1}\ (2^i)^{m} \ }[/math] is prime, then [math]\displaystyle{ m=p^n }[/math].
- [math]\displaystyle{ F_{n,r} }[/math] numbers are pairwise relatively prime within a rank and across ranks: [math]\displaystyle{ gcd(F_{n,i},F_{m,j}) =1 }[/math] for all n, m, i and j.
- They satisfy a product property like Fermat numbers have. And every [math]\displaystyle{ F_{n,r} }[/math] passes Fermat's test to base 2.
Saouter has proven that [math]\displaystyle{ F_{n,2} }[/math] numbers can be proven prime by using Pépin's test with k=5.
Dubner
In 1985, Dubner for the first time built a list of large primes of the form: b2m+1, b ≥ 2 and m ≥ 1.
See also: H.Dubner, W.Keller: "Factors of generalized Fermat numbers" (1995)[1]
Björn & Riesel
In 1998, Björn & Riesel[2] for the first time built a list of large primes of the form: a2m+b2m, b > a ≥ 2 and m ≥ 1.
Notes for this Wiki
Divisibilities of Generalized Fermat numbers for any Proth primes k•2n+1 are listed as GF Divisor on their own page. They are listed as F(n), GF(n,a) or xGF(n,a,b) as used by the output of PFGW or at the The Prime Pages.
To test any Proth prime for divisibilities of Generalized Fermat numbers the program PFGW can be used. For example to test Proth 3•241+1 call
pfgw -gxo -a2 -q"3*2^41+1"
which results in
3*2^41+1 is a Factor of F38!!!! (0.000000 seconds) 3*2^41+1 is a Factor of GF(38,3)!!!! (0.000000 seconds) 3*2^41+1 is a Factor of xGF(37,3,2)!!!! (0.000000 seconds) 3*2^41+1 is a Factor of xGF(38,4,3)!!!! (0.000000 seconds) 3*2^41+1 is a Factor of GF(35,6)!!!! (0.000000 seconds) 3*2^41+1 is a Factor of xGF(36,8,3)!!!! (0.000000 seconds) 3*2^41+1 is a Factor of xGF(38,9,2)!!!! (0.000000 seconds) 3*2^41+1 is a Factor of xGF(38,9,8)!!!! (0.000000 seconds) 3*2^41+1 is a Factor of xGF(39,11,7)!!!! (0.000000 seconds) 3*2^41+1 is a Factor of GF(38,12)!!!! (0.000000 seconds) GFN testing completed
listed on the GF Divisor page.
Only proper factors of Generalized Fermat numbers are listed.
Special conditions for Proth primes
A Proth prime written in the normalized form k•bn+1 is also a Generalized Fermat number under special conditions:
If the k-value is a square and the n-value even the Proth prime k•bn+1 is also a Generalized Fermat number of the form
- [math]\displaystyle{ \Large (\sqrt{k}\cdot b^{n\over 2})^{2^1} + 1 }[/math].
Example: The Proth prime 289•218502+1 is also a Generalized Fermat number of the form (17*29251)21+1. See 289*218502+1 which is also a Cullen prime.
External links
- Generalized Fermat numbers
- Factors of Generalized Fermat numbers found after Björn & Riesel
- Extension with new factors of Generalized Fermat numbers
- Factorizations of small Generalized Fermat numbers
- The smaller Fermat numbers Fm, m<33
- Factors of generalized Fermat numbers found after Björn & Riesel (original, archived copy from September 2007)
- MathWorld article
- Generalized Fermat Prime Search
- List of generalized Fermat primes in bases up to 1000
- List of generalized Fermat primes in bases up to 1030
- Factorization of numbers of the form Fn,2: it includes a program to factor generalized Fermat numbers.
References
- ↑ H.Dubner, W.Keller: "Factors of generalized Fermat numbers" Math. Comp. 64 (1995), 397-405
- ↑ A.Björn, H.Riesel: "Factors of generalized Fermat numbers", Math. Comp. 67 (1998), pp. 441-446
Miscellaneous |
Fermat numbers |
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Gen. Fermat primes |
Gen. Fermat primes |
Gen. Fermat primes categories |
GF Divisors |