2260.21 – A Fourth Degree Equation

Find the number of distinct ordered pairs $(x, y)$ , where $x$ and $y$ are positive integers and satisfy the equation

$x^4 y^4 - 10 x^2 y^2 + 9 = 0.$

Solution

The equation factors as follows:

$0 = x^4 y^4 - 10 x^2 y^2 + 9 = (x^2 y^2 - 9) (x^2 y^2 - 1) = (x y + 3) (x y - 3) (x y + 1) (x y - 1)$

Setting each factor in turn equal to zero we obtain solutions.

$x y = -3$ yields no positive solutions.

$x y = 3$ has two positive integer solutions: (1, 3) and (3, 1).

$x y = -1$ also has no positive solutions.

Finally, $x y = 1$ has a single solution: (1, 1).

Thus there are 3 solutions all told.