Rocky Mountain Journal of Mathematics

Subgroup avoidance for primes dividing the values of a polynomial

Paul Pollack

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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

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

First available in Project Euclid: 21 November 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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


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.

Export citation


  • P.T. Bateman and M.E. Low, Prime numbers in arithmetic progressions with difference $24$, Amer. Math. Month. 72 (1965), 139–143.
  • Mihály Bauer, Über die arithmetische Reihe, J. reine angew. Math. 131 (1906), 265–267.
  • Alfred Brauer, A theorem of M. Bauer, Duke Math. J. 13 (1946), 235–238.
  • J.W.S. Cassels and A. Fr öhlich, eds., Algebraic number theory, Academic Press, Inc., London, 1986.
  • I. Gerst and J. Brillhart, On the prime divisors of polynomials, Amer. Math. Month. 78 (1971), 250–266.
  • E. Landau, Handbuch der Lehre von der Verteilung der Primzahlen, Chelsea Publishing Company, New York, 1953.
  • H.W. Lenstra, Jr., and P. Stevenhagen, Primes of degree one and algebraic cases of \uCebotarev's theorem, Enseign. Math. 37 (1991), 17–30.
  • D.A. Marcus, Number fields, Universitext, Springer-Verlag, New York, 1977.
  • M.R. Murty, Primes in certain arithmetic progressions, J. Madras University 51 (1988), 161–169.
  • M.R. Murty and N. Thain, Prime numbers in certain arithmetic progressions, Funct. Approx. Comm. Math. 35 (2006), 249–259.
  • T. Nagell, Introduction to number theory, Chelsea Publishing Company, New York, 1964.
  • W. Narkiewicz, The development of prime number theory, Springer Mono. Math., Springer-Verlag, Berlin, 2000.
  • I. Schur, Über die Existenz unendlich vieler Primzahlen in einigen speziellen arithmetischen Progressionen, Sitzungsber. Berlin. Math. Ges. 11 (1912), 40–50.
  • A. Selberg, An elementary proof of Dirichlet's theorem about primes in an arithmetic progression, Ann. Math. 50 (1949), 297–304.
  • H.N. Shapiro, On primes in arithmetic progression, II, Ann. Math. 52 (1950), 231–243.
  • J. Wójcik, A refinement of a theorem of Schur on primes in arithmetic progressions, Acta Arith. 11 (1966), 433–436.
  • ––––, A refinement of a theorem of Schur on primes in arithmetic progressions, II, Acta Arith. 12 (1966), 97–109.
  • ––––, A refinement of a theorem of Schur on primes in arithmetic progressions, III, Acta Arith. 15 (1968), 193–197.
  • H. Zassenhaus, Über die Existenz von Primzahlen in arithmetischen progressionen, Comm. Math. Helv. 22 (1949), 232–259.