eng
episciences.org
Discrete Mathematics & Theoretical Computer Science
1365-8050
2007-01-01
Vol. 9 no. 2
10.46298/dmtcs.394
394
journal article
Addition and multiplication of beta-expansions in generalized Tribonacci base
Petr Ambrož
Zuzana Masáková
Edita Pelantová
https://orcid.org/0000-0003-3817-2943
We study properties of β-numeration systems, where β > 1 is the real root of the polynomial x3 - mx2 - x - 1, m ∈ ℕ, m ≥ 1. We consider arithmetic operations on the set of β-integers, i.e., on the set of numbers whose greedy expansion in base β has no fractional part. We show that the number of fractional digits arising under addition of β-integers is at most 5 for m ≥ 3 and 6 for m = 2, whereas under multiplication it is at most 6 for all m ≥ 2. We thus generalize the results known for Tribonacci numeration system, i.e., for m = 1. We summarize the combinatorial properties of infinite words naturally defined by β-integers. We point out the differences between the structure of β-integers in cases m = 1 and m ≥ 2.
https://dmtcs.episciences.org/394/pdf
[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]