Sequences of $\{0,1\}$-polynomials with exponents in arithmetic progression

Abstract

This paper finds the first irreducible polynomial in the sequence $f_1(x)$, $f_2(x), \ldots$, where $f_k(x) = 1 + \sum_{i=0}^k x^{n+id}$, based on the values of $n$ and $d$. In particular, when $d$ and $n$ are distinct, the author proves that if $p$ is the smallest odd prime not dividing $d-n$, then $f_{p-2}(x)$ is irreducible, except in a few special cases. The author also completely characterizes the appearance of the first irreducible polynomial, if any, when $d=n$.