# Square roots in square brackets

**Number Theory**Level 4

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

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 \(\displaystyle \sum _{ i=1 }^{ n }{ { x }_{ i }{ y }_{ i } } \).