It can be proven that multiplicative group of integers modulo $N$ defined as
$$\mathbb{Z}^\times_N = \{ i\in \mathbb Z : 1\leq i\leq N−1\; \text{ and }\; \gcd(i,N)=1 \}$$
is cyclic for a prime $N$ and that if it is of prime order, then every non-identity element in the group is a generator of this group.
How can I find such $N$?
I wrote a simple program and brute-forced for $i \in \mathbb Z; i\in [1, 300 000]$ and found no group of prime order so far.