5. Find all solutions to the equation $2017^{x}-2016^{x}=1$.

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5. It is obvious that $x=1$ will be a solution to the given equation. Let's show that there are no others. We have: $\left(\frac{2017}{2016}\right)^{x}=1+\frac{1}{2016^{x}} ;\left(\frac{2017}{2016}\right)^{x}-1=\frac{1}{2016^{x}}$ From this, it is clear that the function on the left side is increasing, while the function on the right side is decreasing. Therefore, the equation has no more than one solution. Answer: $\{1\}$.