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

Carrie E. Finch

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 $d=n$.

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DOI: http://dx.doi.org/10.7146/math.scand.a-15197

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ISSN 0025-5521 (print) ISSN 1903-1807 (online)

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