Arithmetical Properties of the Leaping Convergents of $e^{1/s}$

Abstract

Let $p_k/q_k=[a_0;a_1,a_2,\cdots,a_k]$ be the $k$-th convergent of the continued fraction expansion of a real number $\alpha$. We shall show several interesting arithmetic properties concerning every third convergent of the continued fraction expansion of $e^{1/s}$ ($s\ge 1$).