Prove that there are no positive integers x and y such that: 1/x^2 + 1/xy + 1/y^2 = 1

We know, from enunciation, that the integer numbers x, y
are positive.

To perform the addition of the fractions from
the left side, we'll calculate the least common
denominator:

LCD =
x^2*y^2

Now, we'll multiply each fraction by the needed
value, in order to get x^2*y^2 at denominator.

y^2/x^2*y^2
+ x*y/x^2*y^2 + x^2/x^2*y^2 = 1

(y^2 + xy + x^2)/x^2*y^2 =
1

We'll cross multiply and we'll
get:

y^2 + xy + x^2 =
x^2*y^2

We'll multiply (x-y) both
sides:

(x-y)(y^2 + xy + x^2) =
(x-y)*x^2*y^2

We'll get to the left a difference of
cubes:

x^3 - y^3 = x^3*y^2 -
x^2*y^3

Assuming that x=y=1 =>
1-1=1-1=0

Assuming that x=y=2 => 8-8 = 8*4 - 4*8 =
0

Assuming that x`!=` y, such as x = 2 and y =
3.

x^3 - y^3 = 8-27 =
-19

x^3*y^2 - x^2*y^3 = 8*9 - 4*27 =
-36

For any x y>0, x,y Z, the given
expression is not an identity, while for integer positive values of x and y, the given
expression does represent an identity.