Generalized Fermat number

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 the Pépin's test with k=5.

Dubner

In 1985, Dubner for the first time built a list of large primes of the form: b^2m+1, b ≥ 2 and m ≥ 1.

Björn & Riesel

In 1998, Björn & Riesel for the first time built a list of large primes of the form: a^2m+b^2m, b > a ≥ 2 and m ≥ 1.

External links

• Generalized Fermat numbers
• Factorization of numbers of the form F_n,2: it includes a program to factor generalized Fermat numbers.
• http://www1.uni-hamburg.de/RRZ/W.Keller/GFNfacs.html Factors 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

• Anders Björn and Hans Riesel, Factors of generalized Fermat numbers, Math. Comp. 67 (1998), pp. 441-446