Question of Number Theory

Firstly, notice that both sides are odd. Therefore

$2n^2+1|n^3+9n-17$

$\iff 2n^2+1|2(n^3+9n-17)$

$\iff 2n^2+1|2n^3+18n-34-n(2n^2+1)=17n-34$ .

The motivation behind this is to clear the cube, which would be $2n^2+1$ multiplied by $n$ , something natural to do. But since the coefficient of $n$ is $1$ , we multiply the dividend expression by $2$ as we observe both sides are odd.

The rest is easy to follow. As $2n^2+1>17n-34$ for $n>5$ by solving the quadratic equation $2n^2-17n+35=0$ . We then have for $n=1,2,3,4,5, \{2n^2+1, 17n-34\}=\{5, -17\}, \{9, 0\}, \{19, 17\}, \{33, 34\}, \{51, 51\}$ , and therefore the only solution are $n=2$ and $n=5$ , with the answer $2+5=\boxed{7}$ .