# Thread: Existence of Primitive Roots

1. ## Existence of Primitive Roots

By definition, the positive integers that have a primitive root are 2, 4 and integers of the form $p^t$ and $2p^t$ where p is a prime and t is a positive integer.

Then why does 16 NOT have a primitive root? $16 = 2^4$ and 2 is prime and 4 is a positive integer.

I'm confused...

2. ## Re: Existence of Primitive Roots

You are missing one key word in your definition. There exists a primitive root (mod n) if and only if n = 1, 2, 4, $p^\alpha, 2p^\alpha$, where p is an odd prime.