Slowly Solve Diophantine Equation x²(y-1)+y²(x-1)=1
=====================================================
I have seen on YouTube how the equation is solved with the
substitution:
1) u=x+1
2) v=y+1
I want to solve it without substitution and to show some thinking.
So, the requested solutions are pairs of integers (x,y) satisfying the equation.
Let us start by expanding:
xy(x + y) - x² - y² = 1
==> xy(x+y) - (x² + y²) = 1
Now, a good way to complete x² + y² to a multiple of x + y is to add 2xy
to it. To be more precise, we need to subtract it and add it:
xy(x + y) + 2xy - (x² + y²) - 2xy ==>
==> xy(x + y) + 2xy - (x²+ 2xy + y²) = 1
==> xy(x + y + 2) - (x + y)(x + y) = 1
Great! Now, let us complete (x+y)(x+y) to a multiple of (x + y + 2):