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Under which condition on $a_0, \dots , a_n$ the continued fraction expansion of a positive rational number $$r= a_0- \frac{1}{ \ddots -\frac{1}{a_n}}$$ is unique?

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  • $\begingroup$ $a_i\geq 2.\phantom{}$ $\endgroup$ Jun 6, 2015 at 14:02
  • $\begingroup$ So, any positive rational number has a unique finite continued fraction expansion with the $a_i \geq 2$? $\endgroup$ Jun 6, 2015 at 14:12
  • $\begingroup$ Any rational number $\geq 1$. That happens since $x=1$ is the only fixed point of the map $x\to 2-\frac{1}{x}$. $\endgroup$ Jun 6, 2015 at 14:30
  • $\begingroup$ But I need something also in the case $0 \leq x \leq 1$ !!! $\endgroup$ Jun 6, 2015 at 15:01
  • $\begingroup$ If you are able to represent any $x\in\mathbb{Q}_{\geq 1}$ as a negative continued fraction and you have a rational number $y\in(0,1)$, you may represent $\frac{1}{y}$, too. $\endgroup$ Jun 6, 2015 at 15:38

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