Rocky Mountain Journal of Mathematics

Subgroup avoidance for primes dividing the values of a polynomial

Paul Pollack

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Abstract

For $f \in \mathbb{Q) [x]$, we say that a rational prime~$p$ is a \textit {prime divisor} of~$f$ if $p$ divides the numerator of $f(n)$ for some integer $n$. Let $\mathcal{P} (f)$ denote the set of prime divisors of~$f$. We present an elementary proof of the following theo\-rem, which generalizes results of Bauer and Brauer: fix a nonzero integer~$g$. Suppose that $f(x) \in \mathbb{Q} [x]$ is a nonconstant polynomial having a root in $\mathbb{Q} _p$ for every prime $p$ dividing $g$, and having a root in $\mathbb{R} $ if $g \lt 0$. Let $m$ be a positive integer coprime to~$g$, and let~$H$ be a subgroup of $(\mathbb{Z} /m\mathbb{Z} )^{\times }$ not containing $g\bmod {m}$. Then there are infinitely many primes $p \in \mathcal{P} (f)$ with $p\bmod {m} \notin H$.

Article information

Source
Rocky Mountain J. Math., Volume 47, Number 6 (2017), 2043-2050.

Dates
First available in Project Euclid: 21 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1511254963

Digital Object Identifier
doi:10.1216/RMJ-2017-47-6-2043

Mathematical Reviews number (MathSciNet)
MR3725255

Zentralblatt MATH identifier
06816581

Subjects
Primary: 11N32: Primes represented by polynomials; other multiplicative structure of polynomial values
Secondary: 11A41: Primes 11N13: Primes in progressions [See also 11B25]

Keywords
Primes in progressions Dirichlet's theorem elementary methods prime divisors of polynomials

Citation

Pollack, Paul. Subgroup avoidance for primes dividing the values of a polynomial. Rocky Mountain J. Math. 47 (2017), no. 6, 2043--2050. doi:10.1216/RMJ-2017-47-6-2043. https://projecteuclid.org/euclid.rmjm/1511254963


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