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September 20th, 2017, 10:24 AM  #1 
Newbie Joined: Sep 2017 From: UK Posts: 2 Thanks: 0  Order of an element in multiplicative group
Hi I am looking for the answer how many possible values has $\displaystyle Ord_{p} (2)$ I suspect that there is only two solutions : $\displaystyle 0.5*(p1)$ and $\displaystyle p1$. Could someone prove it? 
September 20th, 2017, 01:17 PM  #2 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 202 Thanks: 60 Math Focus: Algebraic Number Theory, Arithmetic Geometry 
I'm afraid your suspicion is wrong: $Ord_{31}(2) = 5 < 0.5 * (311) < 31  1$. The subsequent mersenne primes are all also counterexamples. It's easy to see that $Ord_{p}(2) > \log_2(p)$, but I can't think of a straightforward way to improve upon this bound. 
September 20th, 2017, 04:20 PM  #3 
Senior Member Joined: Aug 2012 Posts: 1,956 Thanks: 547 
Found this. https://mathoverflow.net/questions/6...moduloprimes They mentioned your observation that the order is greater than $\log_2(p)$. Evidently not much is known about this. 

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element, group, multiplicative, order 
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