Finite projective planes and a question about primes

Let $n$ be an even integer not divisible by 3. Suppose that $p = {n^2} + n + 1$ is a prime and ${2^{n + 1}} \equiv 1\left( {\bmod p} \right)$. The question is asked whether this can only occur if $n$ is a power of 2. It is noted that an affirmative answer to this question implies that a finite projective plane with a flag transitive collineation group is Desarguesian.