How Many Pairs?

\[ \large \displaystyle x^2+615=2^y\]

Let \(\displaystyle (x_{1},y_{1}),(x_{2},y_{2}),\ldots,(x_{n},y_{n})\) be the integral solution pairs \(\displaystyle (x,y)\) to the equation above such that \(\displaystyle x_1<x_2<x_3<\ldots.<x_n\).

Determine the value of \(\displaystyle \sum\limits_{i=1}^n (-1)^ix_iy_i\).

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