How do solve

NO SOLUTIONS PLEASE, just hints or even just which theorem/method to use.

I thought it might be something to do with primitive roots...

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- Mar 2nd 2010, 02:38 AMDeadstarx^6 congruent to 1 (mod 19)
How do solve

NO SOLUTIONS PLEASE, just hints or even just which theorem/method to use.

I thought it might be something to do with primitive roots... - Mar 2nd 2010, 03:05 AMBacterius
Hello !

Note that is equivalent to . Doesn't this remind you of some algebraïc structure learnt while studying factorization ? :) - Mar 2nd 2010, 03:54 AMtonio
- Mar 2nd 2010, 07:10 AMDeadstar
Thanks for the hint but I probably should have asked about equations of the form with p prime as the above hint for this one hasn't helped me.

Factorizing that I can only see two solutions, 1 and 18.

I can't find anything in our notes about this and it's stuck in the middle of a tutorial consisting of primitive root questions so I don't know how to approach it. - Mar 2nd 2010, 08:24 AMchiph588@
- Mar 2nd 2010, 09:00 AMtonio
- Mar 2nd 2010, 09:46 AMtonio
- Mar 2nd 2010, 12:37 PMDeadstar
EDIT. Solving is happening...

- Mar 2nd 2010, 12:52 PMDeadstar
Ok I can get 7 and 11 from the quadratics but since they both give the same result how do I find 8 and 12?

- Mar 2nd 2010, 06:04 PMBacterius
I think you already worked out what I'm about to say, but this is just to increase my post count (joking :D)

So the solutions of are given by :

Solve those for and you will find all the solutions of the congruence. Don't forget to include the four solutions given by the quadratics ;) - Mar 2nd 2010, 06:54 PMtonio

What "same result" are you talking about? That they have the same discriminant doesn't mean their roots are the same...(Wondering).

Just apply the well-known formula for the roots of a quadratic equation, which you can do since the characteristic of the field we're working on isn't two.

Tonio