
Primitive Root questions
Hi, I have 3 problems concerning primitive roots that I have been unable to work through. Any advice on how to solve these would be greatly appreciated!
1. Show if g, h are primitive roots of p and p is odd, the least residue of gh is not a primitive root of p.
2. Show 131071 = (2^17)1 is prime.
3. Show that k exists such that g^(k+1)=(g^k)+1 (mod p), where g is a primitive root of p and p is a prime.
Thanks in advance!

Some ideas/hints:
1) If g is a primitive root, what is $\displaystyle g^{\tfrac{p1}{2}}(\bmod.p)$
2) Well consider a prime q dividing $\displaystyle n=2^{17}1$ , then the order of 2 mod q is (Wink) ? . But the order of 2 mod q divides $\displaystyle q1$ by Fermat's Little Theorem, this reduces the number of primes we have to check (to see if they divide n) in fact, remember that if n were not prime, there's a prime divisor which is no greater than $\displaystyle \sqrt{n}$.
3) $\displaystyle g^{k+1}g^k\equiv{1}(\bmod.p)$ ...