# primitive roots

1. ### Resources for questions on primitive roots

Does anyone know of any resources on questions on primitive roots and order of a modulo n? They need to be suitable for elementary number theory course. (These could be interesting results and challenging ones).
2. ### What is the most motivating way to introduce primitive roots?

I am teaching elementary number theory to first year undergraduate students. How do introduce the order of an integer modulo n and primitive roots? How do I make this a motivating topic and are there any applications of this area? I am looking at something which will have an impact. These...
3. ### prove that φ(n/p^K) is even

so in this problem i no that gcd(a,n)=1 and the question i need to answer is : let p be any prime factor of n and let k be the number of times that p appears in the prime factorization of n. prove that φ(n/p^K) is even so far i have If p is prime, then φ(p^k) = p^(k-1)(p-1). So, if p is odd...
4. ### p=2^k+1 and (a/p)=-1 imply a is primitive modulo p

Hello! I've been working on this little proof but I never got it. I'm pretty sure it's real simple but I'm having trouble and I'd appreciate it if someone could help me. If p=2^k+1 is prime and (a/p)=-1 modulo p, show that a is a primitive root modulo p. By Lagrange's Theorem, a^{2^k}\equiv...
5. ### Congruence Problem

Hi, I need to solve this congruence. 7^x \equiv 6 (mod17) In the question, it has a squiggly line at the top of the congruence sign, but i'm not sure what that means :(. I know this questions probably has something to do with primitive roots, but I really dont know how to start...
6. ### Simple polynomial problem?

A problem from G.E. Andrews's "Number Theory": Suppose g is a primitive root modulo p (a prime) and m | p-1, where (0<m<p-1). How many integral solutions are there of the congruence x^m - g "congruent to" 0 (mod p). Proposed solution: Assume there is at least one such x...
7. ### Stumped by "simple" polynomial problem

A problem from G.E. Andrews's "Number Theory": Suppose g is a primitive root modulo p (a prime) and m | p-1, where (0<m<p-1). How many integral solutions are there of the congruence x^m - g "congruent to" 0 (mod p). Proposed solution: Assume there is at least one such x...
8. ### Primitive Roots and Quadratic Residues

I'm struggling with the concept of primitive roots and their application in certain proofs. I'm struggling with starting this problem: Let a and n be in the natural numbers and let p be an odd prime where p does not divide a. Using primitive roots show x^2 \equiv a mod p is solvable if and...