According to the Wikipedia article a composite number n is a strong pseudoprime to at most one quarter of all bases below n .
Do Fermat pseudoprimes have some similar property ? Is it known what is the largest number of bases for which composite n , that is not Carmichael number is...
Does exist Fermat pseudoprime $n$ such that $n$ is a pseudoprime for all odd bases in interval :
$\left [3~,~2\cdot \left \lfloor \frac{\sqrt[3]n}{2} \right \rfloor +1 \right]$
Hi ,
I forgot to point out that p has to be greater than three . I know that first two cases are not mutually exclusive . Both values of S_0 can be used in order to prove primality of corresponding Wagstaff number .
type \displaystyle in front of \prod$$
pr(X=k)=\mu\frac{\displaystyle\prod^{k-1}_{i=1}\{1-\mu+(i-1)\theta\}}{\displaystyle\prod^{k}_{i=1}\{1+(i-1)\theta\}}, \quad \mbox{for k$\geq$ 1,}$$
Hi . You don't have to calculate value of S_{2^{n+1}-3} to find out whether F_n(132) \mid S_{2^{n+1}-3} See Wikipedia article : Lucas-Lehmer primality test
One can formulate similar conjectures for other Generalized Fermat numbers .
Primality test based on this conjecture written in...
These polynomials are not cyclotomic polynomials.
f_n can be rewritten into form :
f_n=\displaystyle \sum_{i=0}^n x^{i}+(a-1)\cdot \displaystyle \sum_{i=0}^k x^{i} ,or
f_n=\frac{x^{n+1}+(a-1)x^{k+1}-a}{x-1}
Is it true that polynomials of the form :
f_n= x^n+x^{n-1}+\cdots+x^{k+1}+ax^k+ax^{k-1}+\cdots+a
where \gcd(n+1,k+1)=1 , a\in \mathbb{Z^{+}} , a is odd number , a>1, and a_1\neq 1
are irreducible over the ring of integers \mathbb{Z}...