Proc. Japan Acad. Ser. A Math. Sci. 92 (3), 39-40, (March 2016) DOI: 10.3792/pjaa.92.39
Diego Marques, Carlos Gustavo Moreira
KEYWORDS: Liouville numbers, Diophantine numbers, Continued fraction, 11K60, 11J81
Let $\ell$ be the Liouville’s constant, defined as a decimal with a 1 in each decimal place corresponding to $n!$ and 0 otherwise. This number is a classical example of a Liouville number. In this note, we give an optimal condition on the number of replacements of 0’s by 1’s between two consecutive 1’s in the decimal expansion of $\ell$ in order to ensure that this new number is still a Liouville number.