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CARMICHAEL NUMBERS IN ARITHMETIC PROGRESSIONS

Part of: Multiplicative number theory Elementary number theory

Published online by Cambridge University Press:  08 March 2013

KAISA MATOMÄKI
KAISA MATOMÄKI*
Affiliation:
Department of Mathematics, University of Turku, 20014 Turku, Finland email ksmato@utu.fi
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Abstract

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We prove that when $(a, m)= 1$ and $a$ is a quadratic residue $\hspace{0.167em} \mathrm{mod} \hspace{0.167em} m$, there are infinitely many Carmichael numbers in the arithmetic progression $a\hspace{0.167em} \mathrm{mod} \hspace{0.167em} m$. Indeed the number of them up to $x$ is at least ${x}^{1/ 5} $ when $x$ is large enough (depending on $m$).

Keywords

Carmichael numberarithmetic progression

MSC classification

Secondary: 11N25: Distribution of integers with specified multiplicative constraints 11A51: Factorization; primality

Information

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 94 , Issue 2 , April 2013 , pp. 268 - 275
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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