On a Zaremba’s Conjecture for Powers

A conjecture of Zaremba says that for every m≥ 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· 2ⁿ-th power of $7$ where n≥ 0 and $c=1,3,5,7,9,11$. A more general case is considered. 2000 Mathematics Subject Classification. 11A55