A={x|x=a^2+b^2,a,b∈z}x1,x2∈A求证：x1*x2∈A

Question

A={x|x=a^2+b^2,a,b∈z}
x1,x2∈A
求证：x1*x2∈A

Answer 1

x1∈A,则存在a1、b1∈Z,使得x1＝a1^2＋b1^2
x2∈A,则存在a2、b∈Z,使得x1＝a2^2＋b2^2
∴x1·x2
＝(a1^2＋b1^2)(a2^2＋b2^2)
＝a1^2a2^2＋a1^2b2^2＋a2^2b1^2＋b1^2b2^2
＝a1^2a2^2＋b1^2b2^2＋2a1a2b1b2＋a1^2b2^2＋a2^2b1^2－2a1a2b1b2
＝(a1a2＋b1b2)^2＋(a1b2－a2b1)^2
显然,a1a2＋b1b2∈Z,a1b2－a2b1∈Z
故x1·x2∈A