Sum of Squares Cannot Equal Product of Squares
\[x^2+y^2=x^2y^2\]
has no non trivial integer solutions.Working modulus
\[x^2\]
:\[x^2+y^2 =x^2y^2 \rightarrow y^2 \equiv 0 \; (mod \; x^2)\]
Working modulus
\[y\]
:\[x^2+y^2 =x^2y^2 \rightarrow x^2 \equiv 0 \; (mod \; y^2)\]
Hence
\[x^2 | y^2, \; y^2 | x^2\]
. This means that \[x^2=y^2\]
.Let
\[x^2=y^2\]
then \[2y^2=y^4 \rightarrow y^2(2-y^2)=0 \rightarrow y=0, \pm \sqrt{2}\]
. Hence there are no non trivial integer solutions. 
