Question on irreducible polynomials and primes.

Question

Consider the polynomial $p(x) = 1+\sum_{i=1}^d a_i x^i$ where $a_i$ is binary and not all $a_i$ are $0$. Is it possible that $p(2^n)$ is prime for all integer $n>-1 ?$

Answer 1

No.
Let $q = p(1) = 1 + \sum_{i=1}^d a_i$. Take $n$ so that $2^n \equiv 1 \mod q$ and you have $p(2^n) \equiv 0 \mod q$.