2260.21 – A Fourth Degree Equation

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

x4y4−10x2y2+9=0x^4 y^4 - 10 x^2 y^2 + 9 = 0

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Solution

The equation factors as follows:

0=x4y4−10x2y2+9=(x2y2−9)(x2y2−1)=(xy+3)(xy−3)(xy+1)(xy−1)\begin{align*}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)\end{align*}

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

  • xy=−3x y = -3 yields no positive solutions.

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

  • xy=−1x y = -1 also has no positive solutions.

  • xy=1x y = 1 has a single solution: (1, 1).

Thus there are 3 solutions all told.