[tex]\text{Let} ~ W_p ~ \text{be a Wagstaff number of the form :} W_p = \frac{2^p+1}{3}~, \text{where}~p>3 [/tex](adsbygoogle = window.adsbygoogle || []).push({});

[tex]\text {Let's define }~S_0~ \text{as :}[/tex]

[tex]S_0 =

\begin{cases}

3/2, & \text{if } p \equiv 1 \pmod 4 \\

11/2, & \text{if } p \equiv 1 \pmod 6 \\

27/2, & \text{if} ~p \equiv 11 \pmod {12} ~\text{and}~p \equiv 1,9 \pmod {10} \\

33/2, & \text{if}~ p \equiv 11 \pmod {12} ~\text{and}~p \equiv 3,7 \pmod {10} \\

\end{cases} [/tex]

[tex]\text{Next define sequence}~S_i~\text{as :} [/tex]

[tex]S_i =

\begin{cases}

S_0, & i=0 \\

8S^4_{i-1}-8S^2_{i-1}+1, & i>0

\end{cases}[/tex]

[tex] \text{How to prove following statement :} [/tex]

[tex]\text{Conjecture :}[/tex]

[tex]W_p=\frac{2^p+1}{3}~\text{is a prime iff}~S_{\frac{p-1}{2}} \equiv S_0 \pmod {W_p} [/tex]

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# Primality Criteria for Wagstaff numbers

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