Hi, I am having a tough time with this question.
Let and be primitive roots modulo an odd prime and let be an interger co-prime to p.
Using , prove .
Then show that is a primitive root modulo .
For the first part I know that means that and are either both odd or both even. But I need some help with this question.
Thank You for any help.
Let and be integers, with .
Here's a general theorem: If the order of modulo is , then the order of modulo is , for .
It follows that the values of for which has order modulo are exactly those satisfying , or equivalently ; in other words relatively prime to .
In our problem, the fact that is a primitive root of means that has order modulo .
So, has order exactly for integers satisfying . It's clear that is one of them since and therefore has order modulo , i.e. is a primitive root of .