# Square roots in square brackets

$\large y= \left\lfloor \sqrt { 1 } \right\rfloor +\left\lfloor \sqrt { 2 } \right\rfloor +\left\lfloor \sqrt { 3 } \right\rfloor +\dots +\left\lfloor \sqrt { { x }^{ 2 }-1 } \right\rfloor$

Let $$x$$ and $$y$$ be a pair of prime numbers.

If pairs $$\left\{ { x }_{ 1 },{ y }_{ 1 } \right\} ,\left\{ { x }_{ 2 },{ y }_{ 2 } \right\} ,\left\{ { x }_{ 3 },{ y }_{ 3 } \right\} \dots \left\{ { x }_{ n },{ y }_{ n } \right\}$$ are all of the solutions to the equation above, find the value of $$\displaystyle \sum _{ i=1 }^{ n }{ { x }_{ i }{ y }_{ i } }$$.

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