smc

By the properties of logarithms, we can rearrange the equation to read $x^{2017}=2017x$ with $x={\log }_{b}a$. If $x\ne 0$, we may divide by it and get $x^{2016}=2017$, which implies $x=\pm \text{root}2016\text{of}2017$. Hence, we have $3$ possible values $x$, namely $$x=0,x=2017^{\frac{1}{2016}},\text{and}x=-2017^{\frac{1}{2016}}.$$ Since ${\log }_{b}a=x$ is equivalent to $a=b^x$, each possible value $x$ yields exactly $199$ solutions $(b,a)$, as we can assign $a=b^x$ to each $b=2,3,\dots ,200$. In total, we have $3\cdot 199=\boxed{597}$ solutions.