let

then

then

----------- (1)

and

--------------- (2)

You need to find integers x and y such that (1) and (2) are also integers. Your solution of z1=1+1i with x = 1 and y = 1 is incorrect since (1) gives

.

In fact you can go further and find all the possible solutions to your question.

For x and y integers,

must also be an integer. This equation is a circle centered about the origin and only has 4 solutions (along the x and y axes). x = k, y = 0 and x= -k, y=0 etc...

for

or

,

or

so one of (1) or (2) will be a fraction. So

or

.