Question 1046170
Suppose that THERE ARE positive integers x and y such that 

{{{1/x^2 + 1/(xy) + 1/y^2 = 1}}}.

===> {{{1/x^2 + 2/(xy) + 1/y^2 = 1/(xy) +1 = (xy+1)/(xy)}}}.

<===> {{{(1/x + 1/y)^2 = (xy+1)/(xy)}}}.

===> {{{(( x+y)/(xy))^2 = (xy+1)/(xy)}}}.

===> {{{(x+y)^2/(xy)^2 = (xy+1)/(xy)}}}.

<===> {{{(xy)(xy+1) = (x+y)^2}}}

<===> {{{(xy)^2 + (xy) - (x+y)^2 = 0}}}


===> {{{xy = (-1 +- sqrt( 1+4(x+y)^2 ))/2 }}} 

===> {{{xy = (-1 + sqrt( 1+4(x+y)^2 ))/2 }}} , since xy is a positive integer.


Because of the denominator 2, and the -1 in the numerator, we are forced to say that 

{{{1+4(x+y)^2 }}} is an ODD PERFECT SQUARE.

===>{{{ 1+4(x+y)^2  = (2M+1)^2}}} for some positive integer M.

<===> {{{1+4(x+y)^2  = 4M^2 +4M +1}}}

===> {{{(x+y)^2 = M^2+M = M(M+1)}}},

which says that a perfect square ({{{(x+y)^2}}}) is the product of two consecutive positive integers ( M(M+1) ).  But this is impossible.

Hence, a contradiction.

Therefore there CANNOT be two positive integers such that {{{1/x^2 + 1/(xy) + 1/y^2 = 1}}}.