Since $p$ is a positive integer, then $p \geq 1$ and so $0<\frac{1}{p} \leq 1$. Since $n$ is a positive integer, then $n \geq 1$ and so $n+\frac{1}{p}>1$, which tells us that $0<\frac{1}{n+\frac{1}{p}}<1$. Therefore, $m<m+\frac{1}{n+\frac{1}{p}}<m+1$. Since $m+\frac{1}{n+\frac{1}{p}}=\frac{17}{3}$, which is between 5 and 6, and since $m$ is an integer, then $m=5$. Since $m=5$, then $m+\frac{1}{n+\frac{1}{p}}=\frac{17}{3}$ gives $\frac{1}{n+\frac{1}{p}}=\frac{2}{3}$ or $n+\frac{1}{p}=\frac{3}{2}$. Since $n<n+\frac{1}{p} \leq n+1$ and $n$ is an integer, then $n=1$. Thus, $n+\frac{1}{p}=\frac{3}{2}$ gives $\frac{1}{p}=\frac{1}{2}$, which gives $p=2$. Therefore, $n=1$.