Square roots in square brackets

y=1+2+3++x21 \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 xx and yy be a pair of prime numbers.

If pairs {x1,y1},{x2,y2},{x3,y3}{xn,yn}\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 i=1nxiyi\displaystyle \sum _{ i=1 }^{ n }{ { x }_{ i }{ y }_{ i } } .

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