Question

求：x^2+y^2=2(x+y)+xy 的正整数解

Answer 1

x^2+y^2=2(x+y)+xy =>

2x^2+2y^2=4(x+y)+2xy =>

(x^2-2xy+y^2)+(x^2-4x+4)+(y^2-4y+4)=8

(x-y)^2+(x-2)^2+(y-2)^2=8

(x-2)^2 y无解

x=2 => y=4

x=3 => y无解

x=4 => y=2

故x^2+y^2=2(x+y)+xy 的正整数解为(2,4)或(4,2)