Abstract
Let ζk be a k-th primitive root of unity, m ≥ø(k) + 1 an integer and Φk(X) ∈ ℤ[X] the k-th cyclotomic polynomial. In this paper we show that the pair (-m+ζk,N) is a canonical number system, with N = {0,1,...,|Φk(m)|-1}. Moreover we also discuss whether the two bases -m + ζk and -n + ζk are multiplicatively independent for positive integers m, n and k fixed.
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Acknowledgements
The authors want to thank Shigeki Akiyama (University of Tsukuba) for pointing them to the refinement of Petho [16, Theorem 7.1] and Julien Bernat (Université de Lorraine) for many valuable discussions concerning Cobham’s theorem and its requirements.
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Communicated by B. Czédli
Supported by the Austrian Science Fund (FWF) under the project P 24801-N26.
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Madritsch, M.G., Ziegler, V. An infinite family of multiplicatively independent bases of number systems in cyclotomic number fields. ActaSci.Math. 81, 33–44 (2015). https://doi.org/10.14232/actasm-013-825-5
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DOI: https://doi.org/10.14232/actasm-013-825-5