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

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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.

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