Some kinds of primitive and non-primitive words.

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    • Abstract:
      If the length of a primitive word $$p$$ is equal to the length of another primitive word $$q$$ , then $$p^{n}q^{m}$$ is a primitive word for any $$n,m\ge 1$$ and $$(n,m)\ne (1,1)$$ . This was obtained separately by Tetsuo Moriya in 2008 and Shyr and Yu in 1994. In this paper, we prove that if the length of $$p$$ is divisible by the length of $$q$$ and the length of $$p$$ is less than or equal to $$m$$ times the length of $$q$$ , then $$p^{n}q^{m}$$ is a primitive word for any $$n,m\ge 1$$ and $$(n,m)\ne (1,1)$$ . Then we show that if $$uv,u$$ are non-primitive words and the length of $$u$$ is divisible by the length $$v$$ or one of the length of $$u$$ and $$uv$$ is odd for any two nonempty words $$u$$ and $$v$$ , then $$u$$ is a power of $$v$$ . [ABSTRACT FROM AUTHOR]
    • Abstract:
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