Solving just a Trigonometric Equation!

First, we ought to change the equation in some ways just to make it easy to solve.

$29\sin(x+\theta)=(9y-1)^{2}+29$ where $\theta = \arctan(\dfrac{21}{20})$

Then, we see that the minimum of quadratic equation is equal to the maximum of the trigonometric function. So, these two function will be equal if and only if $y=\dfrac{1}{9}$ and $x+\theta = (2n+\dfrac{1}{2})\pi$

When $x=2015$ , $n_{max}=320$

When $x=-2015$ , $n_{min}=-321$

Hence, the numbers of order pair is $N((x,y))=320+321+1=\boxed{642}$