smc

First, let's write $F_1$ as a proper fraction in base $R_1$. To do that, note that: $F_1=0.373737\dots$ Multiplying this equation on both sides $R_1^2$, we get: $R_1^2{F}_1=37.373737\dots$ Subtracting the first equation from the second one, we get: $$\begin{gathered} R_1^2{F}_1-F_1=37 \\ {F}_1(R_1^2-1)=37 \\ {F}_1=\frac{3R_1+7}{R_1^2-1} \end{gathered}$$ Using a very similar method as above, we can see that: $F_1=\frac{3R_1+7}{R_1^2-1}=\frac{2R_2+5}{R_2^2-1}$ and $F_2=\frac{7R_1+3}{R_1^2-1}=\frac{5R_2+2}{R_2^2-1}$. Dividing the 2 equations to get some potential solutions, we get (note that we don't have to worry about division by zero because $R_1,R_2>7$): $$\begin{gathered} \frac{3F_1+7}{7F_1+3}=\frac{2F_2+5}{5F_2+2} \\ 15F_1{F}_2+6F_1+35F_2+14=14F_1{F}_2+35F_1+6F_2+15 \\ {F}_1{F}_2-29F_1+29F_2=1 \\ \left(F_1+29\right)\left(F_2-29\right)=-840 \end{gathered}$$ Since non-integer bases are rarely used, we can try to assume that the solution is positive integers and see if we get any solutions. Notice that $(F_2-29)$ has to be a negative factor of 840. We need to plug in values of $F_2>7$. 840 divides -21, so we plug in 8 to check. Luckily, when $F_2=8$, we see that $F_1=11$, and furthermore, 11+8=19 is one of the answers. We can quickly test it into the original equations $F_1=\frac{3R_1+7}{R_1^2-1}=\frac{2R_2+5}{R_2^2-1}$ and $F_2=\frac{7R_1+3}{R_1^2-1}=\frac{5R_2+2}{R_2^2-1}$, and we see they indeed work. Therefore the answer is $\boxed{19}$