On a Zaremba’s Conjecture for Powers - Komatsu, Takao (Sarajevo Journal of Mathematics, 2024)
A conjecture of Zaremba says that for every $m\ge 2$ there exists a reduced fraction $a/m$ such that its simple continued fraction has all its partial quotients bounded by $5$. This conjecture is proved with all partial quotients bounded by $3$ for $m$ being ${c· 2^n}$-th power of $7$ where $n\ge 0$ and $c=1,3,5,7,9,11$. A more general case is considered. 2000 Mathematics Subject Classification. 11A55