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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 ≥ 2'' and ''m ≥ 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 ≥ 2'' and ''m ≥ 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 ≥ 2'' and ''m ≥ 1''. | + | 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 ≥ 2'' and ''m ≥ 1''. |
==External links== | ==External links== | ||
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==References== | ==References== | ||
| − | + | <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:
- 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.
External links
- Generalized Fermat numbers
- Factorization of numbers of the form Fn,2: it includes a program to factor generalized Fermat numbers.
http://www1.uni-hamburg.de/RRZ/W.Keller/GFNfacs.htmlFactors of generalized Fermat numbers found after Björn & Riesel] (not available anymore)- Factors of generalized Fermat numbers found after Björn & Riesel (original)
- 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
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
