Solving an equation and finding a couple of results given that they are integer

Question

My problem is to find all solutions to an equation with 2 variables between two values that define the range of my search. e.g., $(x+y-y²)/(1-y-x²)=0$ with solutions ${x,y} \in \mathbb{Z}$.

Answer 1

The following might be of use:

Reduce[x + y == y^2 && x^2 + y != 1, {x, y}, Integers]

To ease things, we consider the numerator and denominator separately. Here, we are asking Mathematica for conditions on integer x and y such that the numerator is zero, and the denominator isn't.

Solve[x + y == y^2 && x^2 + y != 1, {x, y}, Integers]

is useful as well.

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If you need examples, you can use

FindInstance[x + y == y^2 && x^2 + y != 1, {x, y}, Integers, 10]

Change 10 to the number of examples you wish to generate.

Answer 2

You can obtain all integer solutions with Reduce:

Reduce[(x + y - y^2)/(1 - y - x^2) == 0, {x, y}, Integers]

which gives

If you also want to restrict to solutions satisfying $x<15$ and $y<42$ then this works

Reduce[(x + y - y^2)/(1 - y - x^2) == 0 && x < 15 && x < 42, {x,y}, Integers]