[FONT='Times New Roman','serif']Let r be a primitive root of the odd prime p . Prove the following.[/FONT]
[FONT='Times New Roman','serif']If p is congruent to 3(mod 4), then -r has order (p-1)/2 modulo p.[/FONT]
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Thus, it cannot be because the order of is .
Thus, .
This tells us, because .
Now we need to prove that if is order of then .
By the above result we see that , since is odd it follows that must be odd.
But then, .
Also, .
Using properties of primitive roots and orders it means .
Looking at your reply to the question in this thread, I have a question very similiar to the one posted originally and am sure that you would be able to help me out also quite simply.
Let r be a primitive root of the odd prime p. If p=1(mod 4), then -r is also a primitive root of p.
I would appreciate the help on this question, I am sure there will be a similar question on my final next week.