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# Generalized Fermat number

There are different kinds of generalized Fermat numbers.

## John Cosgrave

John Cosgrave has studied the following numbers:

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

$\displaystyle{ F_{0,r} }$ generates the Mersenne numbers.
$\displaystyle{ F_{n,1} }$ generates the Fermat numbers.
$\displaystyle{ F_{n,2} }$ generates the Saouter numbers.

Cosgrave has proven the following properties:

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

Saouter has proven that $\displaystyle{ F_{n,2} }$ 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: b2m+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: a2m+b2m, b > a ≥ 2 and m ≥ 1.