Let $n$ and $n$ are positive integers, $b>1$.
Express $n$ in $b$-basis
$$n = a_kb^k + \cdots + a_1b + a_0.$$
We consider the polynomial
$$f_{b,n}(X) = a_kX^k + \cdots + a_1X + a_0 \in \mathbb{Z}[X].$$

Question 1: Let $p$ is a prime number. Then, is it true that $f_{2,p}(X)$ is an irreducible polynomial?
I have checked this question for all $p < 10^6$, and some cases.

Question 2: Is it true that: p is prime number iff $f_{b, p}(X)$ is an irreducible polynomial?
It is clear that if $n$ is a composite number and $b$ is a least prime divisor of $n$, then $f_{b, n}(X)$ is reducible.