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Difference between revisions of "Generalized Fermat number"

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==Dubner==
 
==Dubner==
 
In 1985, Dubner for the first time built a list of large primes of the form: b<sup>2<sup>m</sup></sup>+1, ''b &ge; 2'' and ''m &ge; 1''.
 
In 1985, Dubner for the first time built a list of large primes of the form: b<sup>2<sup>m</sup></sup>+1, ''b &ge; 2'' and ''m &ge; 1''.
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See also: H.Dubner, W.Keller: "Factors of generalized Fermat numbers" (1995)<ref>[https://www.ams.org/journals/mcom/1995-64-209/S0025-5718-1995-1270618-1/ H.Dubner, W.Keller: "Factors of generalized Fermat numbers"] ''Math. Comp.'' 64 (1995), 397-405</ref>
  
 
==Björn & Riesel==
 
==Björn & Riesel==
In 1998, Björn & Riesel for the first time built a list of large primes of the form: a<sup>2<sup>m</sup></sup>+b<sup>2<sup>m</sup></sup>, ''b > a &ge; 2'' and ''m &ge; 1''.
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In 1998, Björn & Riesel<ref>[https://www.ams.org/journals/mcom/1998-67-221/S0025-5718-98-00891-6/ A.Björn, H.Riesel: "Factors of generalized Fermat numbers"], ''Math. Comp.'' 67 (1998), pp. 441-446</ref> for the first time built a list of large primes of the form: a<sup>2<sup>m</sup></sup>+b<sup>2<sup>m</sup></sup>, ''b > a &ge; 2'' and ''m &ge; 1''.
  
 
==External links==
 
==External links==
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==References==
 
==References==
*Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, ''Math. Comp.'' 67 (1998), pp. 441-446
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<references />
 
[[Category:Number]]
 
[[Category:Number]]

Revision as of 07:56, 22 August 2019

There are different kinds of generalized Fermat numbers.

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:

  1. If number [math]\displaystyle{ \sum_{i=0}^{p-1}\ (2^i)^{m} \ }[/math] is prime, then [math]\displaystyle{ m=p^n }[/math].
  2. [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.
  3. 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.

External links

References