This problem is equivalent to showing that $$x^2+5x+16 = 169k\quad\Leftrightarrow\quad x^2 + 5x + (16-169k)=0$$ has no integer solution. If this equation has integer solution then its determinant $$25-4(16-169k) = 169\times 4k -39$$ has to be a square. However it is divisible by $13$ not $169$, thus impossible. Therefore, it cannot be a square which creates a contradiction.