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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 for all $b>1$?

It is clear that if $n$ is a composite number and $b$ is the least prime divisor of $n$, then $f_{b, n}(X)$ is reducible.

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up vote 11 down vote accepted

The answer is yes to both questions: it is a theorem of Brillhart, Filaseta, and Odlyzko (see corollary 2, p.1058):

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