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Old 2020-03-18, 17:19   #50
paulunderwood
 
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Sep 2002
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Quote:
Originally Posted by BrainStone View Post
Is it possible that \(\exists S \left( n \right) \equiv 1 \mod M_p\), where
\(S \left( 1 \right) = 4 \\
S \left( n + 1 \right) = {S \left( n \right)}^2 - 2 \\
M_p = 2^p - 1 \\
n < p, n \in \mathbb{N}, p \in \mathbb{P}\)
M_2 = 3: S_1 is 4 == 1 mod 3.

If S_k^2-2 == 1 mod M_p would imply S_k^2 == 3 mod M_p, but 3 is a QNR over M_p for all prime p>2. Proof?

Last fiddled with by paulunderwood on 2020-03-18 at 17:23
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