\input zb-basic \input zb-ioport \iteman{io-port 06110229} \itemau{L\'opez, Josefina; Stoll, Peter} \itemti{The 2-adic, binary and decimal periods of $1/3^k$ approach full complexity for increasing $k$.} \itemso{Integers 12, No. 5, 907-928 (2012).} \itemab Summary: An infinite word x over an alphabet with b letters has full complexity if for each $m\in \Bbb N$ all the $b^{m}$ words of length m are factors of x. We prove that the periods of $\pm 1/3^k$ in the 2-adic expansion approach full complexity for increasing k: For any $m\in \Bbb N$, the periods for $k>\lceil (m+1)\ln (2)/\ln (3)\rceil$ have complexity $2^m$. Amazingly, these $2^m$ words occur in the period almost the same number of times. On the way, first we prove the same for the binary period. We get a similar result for the decimal period of $1/3 ^{ k}$. \itemrv{~} \itemcc{} \itemut{period of $1/3^k$; 2-adic; binary; decimal; complexity function of a word; frequency of factors; periodicity conjecture; $3x+1$} \itemli{doi:10.1515/integers-2012-0013} \end