Problems need HardWork #23
Number Theory
Level
4
Find the number of integer pairs such that :
8
10
0
29
16
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1 solution
Moderator note:
What motivated the factorization?
( x 2 + 1 ) ( y 2 + 1 ) + 2 ( x − y ) ( 1 − x y ) = 4 ( 1 + x y )
x 2 y 2 − 2 x y + 1 + x 2 + y 2 − 2 x y + 2 ( x − y ) ( 1 − x y ) = 4
( x y − 1 ) 2 + ( x − y ) 2 − 2 ( x − y ) ( x y − 1 ) = 4
( ( x y − 1 ) − ( x − y ) ) 2 = 4
( x + 1 ) ( y − 1 ) = ± 2
If ( x + 1 ) ( y − 1 ) = 2 then
{ x + 1 = 2 y − 1 = 1
{ x + 1 = − 2 y − 1 = − 1
{ x + 1 = 1 y − 1 = 2
{ x + 1 = − 1 y − 1 = − 2
yielding the solutions ( 1 , 2 ) , ( − 3 , 0 ) , ( 0 , 3 ) , ( − 2 , − 1 )
If ( x + 1 ) ( y − 1 ) = − 2 then
{ x + 1 = 2 y − 1 = − 1
{ x + 1 = − 2 y − 1 = 1
{ x + 1 = 1 y − 1 = − 2
{ x + 1 = − 1 y − 1 = 2
yielding the solutions ( 1 , 0 ) , ( − 3 , 2 ) , ( 0 , − 1 ) , ( − 2 , 3 )
So there are 8 pairs which satisfy the given equation