Recently, Morier-Genoud and Ovsienko introduced a $q$-deformation of rational numbers. More precisely, for an irreducible fraction $\frac{ r}s>0$, they constructed coprime polynomials $\mathcal{ R}_{ \frac{ r}s}(q),~ \mathcal{ S}_{ \frac{ r}s}(q) \in { \mathbb Z}[q]$ with $\mathcal{ R}_{ \frac{ r}s}(1)=r,~\mathcal{ S}_{ \frac{ r}s}(1)=s$. Their theory has a rich background and many applications. By definition, if $r \equiv r' \pmod{ s}$, then $\mathcal{ S}_{ \frac{ r}s}(q)=\mathcal{ S}_{ \frac{ r'}s}(q)$. We show that $rr'{ \equiv} -1 \pmod{ s}$ implies $\mathcal{ S}_{ \frac{ r}s}(q)=\mathcal{ S}_{ \frac{ r'}s}(q)$, and it is conjectured that the converse holds if $s$ is prime (and $r \not \equiv r' \pmod{ s}$). We also show that $s$ is a multiple of 3 (resp. 4) if and only if $\mathcal{ S}_{ \frac{ r}s}(\zeta)=0$ for $\zeta=(-1+\sqrt{ -3})/2$ (resp. $\zeta=i$). We give applications to the representation theory of quivers of type $A$ and the Jones polynomials of rational links.
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