Inspired From Other

Number Theory Level 4

Find the number of ordered pairs of whole numbers $(x,y)$ satisfying $(xy-7)^2 =x^2 +y^2$ .

If all these solutions can be expressed as $(x_1, y_1) , (x_2 , y_2) , \ldots , (x_i , y_i)$ , find

$x_1 + y_1 + x_2 + y_2 + \cdots + x_i + y_i .$

1 solution

Shauryam Akhoury
Sep 25, 2018

${ (xy-7) }^{ 2 }={ x }^{ 2 }+{ y }^{ 2 }$

Can be rearranged as

${ (xy-6) }^{ 2 }+13={ (x+y) }^{ 2 }$

Let

${ (xy-6) }^{ 2 }={ a }^{ 2 }$

and

${ (x+y) }^{ 2 }={ b }^{ 2 }$

So,

${ a }^{ 2 }+13={ b }^{ 2 }$

This has integral solutions,

$a=\pm 6$ and $b=\pm 7$

Meaning,

$xy=6\pm 6$

and

$x+y=\pm 7$

These two have 4 whole number solutions,namely,

$(0,7),(3,4),(4,3),(7,0)$

Sum of all the solutions is $28$