Find all integer roots of: $x^2(y-1)+y^2(x-1)=1$

Question

Find all integer roots of: $x^2(y-1)+y^2(x-1)=1$

Obviously $(2,1)$ and $(1,2)$ are two answers. But I was unable to manipulate the equation algebraically giving a useful form for finding all other possible answers!
I also tried to view it as a quadratic equation in terms of $x$, but forming the Delta of that equation didn't help much other than if $y=1$ then certainly Delta would be a square and equation would have answer for $x$...

Answer 1

Thanks to Piquito for pointing out the negative solution.

As stated in the comments, yours are the only positive integer solutions because you'd get $LHS>1$ with larger $x,y.$ Bounding the discriminant $\Delta_x$ (resp. $\Delta_y$) with squares of polynomials of $x$ (resp. $y$) is a natural and, in this case, not a bad idea, in order to show there are no negative solutions.

WLOG let us consider the problem as a quadratic equation in $y$. We have $\Delta_x=x^4+4x^3-4x^2+4x-4,$ and since $(x^2+2x)^2=x^4+4x^3+4x^2$ we may want to try finding the smallest $m,n\in\mathbb{Z^+}$ such that for some $a$ and all $x \le a$ $$(x^2+2x-m)^2<\Delta_x<(x^2+2x-n)^2.$$ It turns out $(m,n)=(8,4)$ works with $a=-6$, and no integer $x$ fulfills $$\Delta_{x}=(x^2+2x-k)^2 $$ for any $k=5,6,7.$ Therefore we only need to check $x=-1,-2,-3,-4,-5$ back in the original equation to see if there are other solutions besides yours: and there are, namely $(x,y)=(-5,2),(2,-5).$

Answer 2

Any point on the curve $x^2(y-1)+y^2(x-1)=1$ is either close to the line $x=1$, to the line $y=1$ or to the line $x+y=-2$, since:

$$x^2(y-1)+y^2(x-1)-1 = (x-1)(y-1)(x+y+2)+(x+y-3)$$

$\hspace{1.5in}$

These lines are asymptotes and the algebraic curve intersects them only at the solutions you already found. With a crude bound, we just have to test every point in the range $[-6,6]^2$ to be sure there are no other integer solutions.

Answer 3

It is easy to verify that $$x^2(y-1)+y^2(x-1)=1\iff2(xy)^3+(x^2+y^2)(xy)^2=(x^2+y^2+1)^2$$ It follows that $xy$ divides $x^2+y^2+1$, in other words one has $$\frac xy+\frac yx+\frac{1}{xy}\in \mathbb Z$$ The obvious solution $x=y=1$ is not solution of the proposed equation. On the other hand the number $3$ given by this $(x,y)=(1,1)$ is given also by $(x,y)=(1,2),(-5,2)$ (as expected, small multiples of $2$ and $5$ are "candidates"). It is not hard to verify that these two (and the corresponding symmetrics respect to the line $y=x$, of course) no other solution exists.

Since O.P.'s equation is symmetric respect to the line $y=x$ then the only solutions are $$(x,y)=(1,2),(2,1),(-5,2),(2,-5)$$