Positivity in coefficient-free rank two cluster algebras

Let $b,c$ be positive integers, $x_1,x_2$ be indeterminates over $\mathbb{Z}$ and $x_m,m\in \mathbb{Z}$ be rational functions defined by $x_{m-1}{x}_{m+1}=x_m^b+1$ if $m$ is odd and $x_{m-1}{x}_{m+1}=x_m^c+1$ if $m$ is even. In this short note, we prove that for any $m,k\in \mathbb{Z}$, $x_k$ can be expressed as a substraction-free Laurent polynomial in $\Z[x_m^{\pm 1},x_{m+1}^{\pm 1}]$. This proves Fomin-Zelevinsky’s positivity conjecture for coefficient-free rank two cluster algebras.