Generalized Fermat number

There are different kinds of generalized Fermat numbers.

John Cosgrave

John Cosgrave has studied the following numbers:

Numbers of the form: $F_{n,r}=\sum _{i=0}^{p-1}{2}^{i{p}^n}=2^{(p-1)p^n}+2^{(p-2)p^n}+...+2^{2p^n}+2^{p^n}+1=(2^{p^{n+1}}-1)/(2^{p^n}-1)$ 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.

$F_{0,r}$ generates the Mersenne numbers.

$F_{n,1}$ generates the Fermat numbers.

$F_{n,2}$ generates the Saouter numbers.

Cosgrave has proven the following properties:

1. If number $\sum _{i=0}^{p-1}(2^i{)}^m$ is prime, then $m=p^n$.
2. $F_{n,r}$ numbers are pairwise relatively prime within a rank and across ranks: $gcd(F_{n,i},F_{m,j})=1$ for all n, m, i and j.
3. They satisfy a product property like Fermat numbers have. And every $F_{n,r}$ passes Fermat's test to base 2.

Saouter has proven that $F_{n,2}$ numbers can be proven prime by using the [Pepin's test] with k=5.

Dubner

In 1985, Dubner for the first time built a list of large primes of the form: $b^{2^m}+1,b>=2,m>=1$.

Björn & Riesel

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

External links

• 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