Abstract. Let $\zeta_k$ be a $k$-th primitive root of unity, $m\geq\phi (k)+1$ an integer and $\Phi_k(X)\in Z [X]$ the $k$-th cyclotomic polynomial. In this paper we show that the pair $(-m+\zeta_k,{\cal N})$ is a canonical number system, with ${\cal N}=\{0,1,\dots,|\Phi_k(m)|-1\}$. Moreover we also discuss whether the two bases $-m+\zeta_k$ and $-n+\zeta_k$ are multiplicatively independent for positive integers $m$, $n$ and $k$ fixed.

DOI: 10.14232/actasm-013-825-5

AMS Subject Classification (1991): 11A63, 11D61, 11D41

Keyword(s): canonical number systems, radix representations, diophantine equations, Nagell--Ljunggren equation

Received November 5, 2013, and in revised form August 1, 2014. (Registered under 75/2013.)