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\begin{abstract}
A prime number $p$ is called $b${\it-elite} if only finitely many generalized Fermat
numbers $F_{b,n}=b^{2^n}+1$ are quadratic residues modulo $p$. Let $p$ be a prime. Write $p-1=2^rh$
with $r\geq 0$ and $h$ odd. Define the length of the \textit{b-Fermat period of $p$} to be the minimal natural
number $L$ such that $F_{b,r+L}\equiv F_{b,r} \,(\bmod\, p).$ Recently M\"uller and Reinhart derived three
conjectures on $b${\it-elite} primes, two of them being the following. (1) For every natural number $b>1$ there
is a $b${\it-elite} prime. (2) There are generalized elite primes with elite periods of arbitrarily large lengths. We
extend M\"uller and Reinhart's observations and computational results to further support above two conjectures. We
show that Conjecture 1 is true for $b\leq10^{13}$ and that for every possible length
$1\leq L\leq40$ there actually exists a generalized elite prime with elite period length $L$.
\end{abstract}
\end{document}%Wednesday, April 8, 2009 at 15:47