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CARMICHAEL NUMBERS IN ARITHMETIC PROGRESSIONS

Part of: Elementary number theory Multiplicative 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 [email protected]
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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
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. 

References

Alford, W. R., Granville, A. and Pomerance, C., ‘On the difficulty of finding reliable witnesses’, in: Algorithmic Number Theory (Ithaca, NY, 1994), Lecture Notes in Computer Science, 877 (Springer, Berlin, 1994), 1–16.Google Scholar
Alford, W. R., Granville, A. and Pomerance, C., ‘There are infinitely many Carmichael numbers’, Ann. of Math. (2) 139 (3) (1994), 703–722.CrossRefGoogle Scholar
Baker, R. C. and Harman, G., ‘Shifted primes without large prime factors’, Acta Arith. 83 (4) (1998), 331–361.CrossRefGoogle Scholar
Baker, R. C. and Schmidt, W. M., ‘Diophantine problems in variables restricted to the values 0 and 1’, J. Number Theory 12 (4) (1980), 460–486.CrossRefGoogle Scholar
Banks, W. D. and Pomerance, C., ‘On Carmichael numbers in arithmetic progressions’, J. Aust. Math. Soc. 88 (3) (2010), 313–321.CrossRefGoogle Scholar
Ekstrom, A., ‘On the infinitude of elliptic Carmichael numbers’, PhD Thesis, University of Arizona, 1999.Google Scholar
Ekstrom, A., Pomerance, C. and Thakur, D. S., ‘Infinitude of elliptic Carmichael numbers’, J. Aust. Math. Soc. 92 (1) (2012), 45–60.CrossRefGoogle Scholar
van Emde Boas, P. and Kruyswijk, D., ‘A combinatorial problem on finite Abelian groups III’, Afd. Zuivere Wisk., 1969-008, Math. Centrum, Amsterdam.Google Scholar
Erdös, P., ‘On pseudoprimes and Carmichael numbers’, Publ. Math. Debrecen 4 (1956), 201–206.CrossRefGoogle Scholar
Harman, G., ‘On the number of Carmichael numbers up to $x$’, Bull. Lond. Math. Soc. 37 (5) (2005), 641–650.CrossRefGoogle Scholar
Harman, G., ‘Watt’s mean value theorem and Carmichael numbers’, Int. J. Number Theory 4 (2) (2008), 241–248.CrossRefGoogle Scholar
Meshulam, R., ‘An uncertainty inequality and zero subsums’, Discrete Math. 84 (2) (1990), 197–200.CrossRefGoogle Scholar
Xylouris, T., ‘On the least prime in an arithmetic progression and estimates for the zeros of Dirichlet $L$-functions’, Acta Arith. 150 (1) (2011), 65–91.CrossRefGoogle Scholar