EVEN AND ODD INTEGRAL PARTS OF POWERS OF A REAL NUMBER
Glasgow mathematical journal, Tome 48 (2006) no. 2, pp. 331-336
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We define a subset $\mathcal Z$ of $(1,+\infty)$ with the property that for each $\alpha \in {\mathcal Z}$ there is a nonzero real number $\xi = \xi(\alpha)$ such that the integral parts $[\xi \alpha^n]$ are even for all $n \in \mathbb{N}$. A result of Tijdeman implies that each number greater than or equal to 3 belongs to $\mathcal{Z}$. However, Mahler's question on whether the number 3/2 belongs to $\mathcal{Z}$ or not remains open. We prove that the set ${\mathcal S}:=(1,+\infty) \textbackslash {\mathcal Z}$ is nonempty and find explicitly some numbers in ${\mathcal Z} \cap$ (5/4,3) and in ${\mathcal S} \cap (1,2)$.
DUBICKAS, ARTŪRAS. EVEN AND ODD INTEGRAL PARTS OF POWERS OF A REAL NUMBER. Glasgow mathematical journal, Tome 48 (2006) no. 2, pp. 331-336. doi: 10.1017/S0017089506003090
@article{10_1017_S0017089506003090,
author = {DUBICKAS, ART\={U}RAS},
title = {EVEN {AND} {ODD} {INTEGRAL} {PARTS} {OF} {POWERS} {OF} {A} {REAL} {NUMBER}},
journal = {Glasgow mathematical journal},
pages = {331--336},
year = {2006},
volume = {48},
number = {2},
doi = {10.1017/S0017089506003090},
url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089506003090/}
}
TY - JOUR AU - DUBICKAS, ARTŪRAS TI - EVEN AND ODD INTEGRAL PARTS OF POWERS OF A REAL NUMBER JO - Glasgow mathematical journal PY - 2006 SP - 331 EP - 336 VL - 48 IS - 2 UR - http://geodesic.mathdoc.fr/articles/10.1017/S0017089506003090/ DO - 10.1017/S0017089506003090 ID - 10_1017_S0017089506003090 ER -
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