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Carmichael meets Chebotarev

  Published:2012-09-21
 Printed: Dec 2013
  • William D. Banks,
    Department of Mathematics, University of Missouri, Columbia, MO 65211 USA
  • Ahmet M. Güloğlu,
    Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, TURKEY
  • Aaron M. Yeager,
    Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
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Abstract

For any finite Galois extension $K$ of $\mathbb Q$ and any conjugacy class $C$ in $\operatorname {Gal}(K/\mathbb Q)$, we show that there exist infinitely many Carmichael numbers composed solely of primes for which the associated class of Frobenius automorphisms is $C$. This result implies that for every natural number $n$ there are infinitely many Carmichael numbers of the form $a^2+nb^2$ with $a,b\in\mathbb Z $.
Keywords: Carmichael numbers, Chebotarev density theorem Carmichael numbers, Chebotarev density theorem
MSC Classifications: 11N25, 11R45 show english descriptions Distribution of integers with specified multiplicative constraints
Density theorems
11N25 - Distribution of integers with specified multiplicative constraints
11R45 - Density theorems
 

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