Show that, for n>1, 3 is a primitive root of any prime of the form . Thoughts, Need conditions such that if 3 is NOT a primitive root then such and such. But I don't know what the such and such should be.
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Originally Posted by Deadstar Show that, for n>1, 3 is a primitive root of any prime of the form . Thoughts, Need conditions such that if 3 is NOT a primitive root then such and such. But I don't know what the such and such should be. In this case, , so for some . Let and look at : . Note that I'm saying since if then would not be prime. So in summary we have and I claim this forces . Can you see why? Thus which makes a primitive root.
Originally Posted by chiph588@ So in summary we have and I claim this forces . Can you see why? Honestly, no.
Originally Posted by chiph588@ So in summary we have and I claim this forces . Can you see why? I'm saying if , then . Let's prove the contrapositive: If for some , then . Well . Thus our original claim is true.
Originally Posted by chiph588@ I'm saying if , then . Let's prove the contrapositive: If for some , then . Well . Thus our original claim is true. Oh ok I see! I was wrongly thinking that it was either going to be to -1 or 1 for some reason. So hence we could have said that if... mod p Then we can square both sides and get a contradiction. But of course can be congruent to more than 1 or -1... Cheers!
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