Diophantines!

Number Theory Level 3

Find the number of ordered pairs of integers ( x , y ) (x, y) satisfying the equation

( x 2 + 1 ) ( y 2 + 1 ) + 2 ( x − y ) ( 1 − x y ) = 4 ( 1 + x y ) \large\ (x^2 + 1)(y^2 + 1) + 2(x - y)(1 - xy) = 4(1 + xy)


The answer is 8.

Priyanshu Mishra by Priyanshu Mishra , 20, India

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1 solution

Chris Lewis
Oct 2, 2019

Expanding the whole expression and collecting terms, we get

x 2 y 2 − 2 x 2 y + x 2 + 2 x y 2 − 4 x y + 2 x + y 2 − 2 y − 3 = 0 x^2 y^2 - 2 x^2 y + x^2 + 2 x y^2 - 4 x y + 2 x + y^2 - 2 y - 3 = 0

This factorises! We get ( x + 1 ) 2 ( y − 1 ) 2 = 4 (x+1)^2 (y-1)^2=4 .

So we either have ( x + 1 ) 2 = 4 (x+1)^2=4 and ( y − 1 ) 2 = 1 (y-1)^2=1 , with a total of 4 4 solutions, or ( x + 1 ) 2 = 1 (x+1)^2=1 and ( y − 1 ) 2 = 4 (y-1)^2=4 , again with a total of 4 4 solutions. So overall there are 8 \boxed8 solutions in integers.

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