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A morphism $f$ is $k$-power-free if and only if $f\left(w\right)$ is $k$-power-free whenever $w$ is a $k$-power-free word. A morphism $f$ is $k$-power-free up to $m$ if and only if $f\left(w\right)$ is $k$-power-free whenever $w$ is a $k$-power-free word of length at most $m$. Given an integer $k\ge 2$, we prove that a binary morphism is $k$-power-free if and only if it is $k$-power-free up to $k^2$. This bound becomes linear for primitive morphisms: a binary primitive morphism is $k$-power-free if and only if it is $k$-power-free up to $2k+1$

A morphism is -power-free if and only if is -power-free whenever is a -power-free word. A morphism is -power-free up to if and only if is -power-free whenever is a -power-free word of length at most . Given an integer ≥ 2, we prove that a binary morphism is -power-free if and only if it is -power-free up to . This bound becomes linear for primitive morphisms: a binary primitive morphism is -power-free if and only if it is -power-free up to