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Carmichael Meets Chebotarev

Published online by Cambridge University Press:  20 November 2018

William D. Banks,
Ahmet M. Güloğlu and
Aaron M. Yeager
William D. Banks
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211 USA e-mail: bankswd@missouri.edu
Ahmet M. Güloğlu
Affiliation:
Department of Mathematics, Bilkent University, 06800 Bilkent, Ankara, Turkey e-mail: guloglua@fen.bilkent.edu.tr
Aaron M. Yeager
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, USA e-mail: amydm6@mail.missouri.edu
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Abstract.

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For any finite Galois extension $K$ of $\mathbb{Q}$ and any conjugacy class $C$ in $\text{Gal}\left( {K}/{\mathbb{Q}}\; \right)$ , 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 ${{a}^{2}}\,+\,n{{b}^{2}}$ with $a,\,b\,\in \,\mathbb{Z}$ .

Keywords

11N2511R45Carmichael numbersChebotarev density theorem

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 4 , 01 December 2013, pp. 695 - 708
Copyright
Copyright © Canadian Mathematical Society 2013

References

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