How many complex numbers $z=x+iy$ are there such that $x+y=1$ and $e^{i(x^2+y^2)}=1.$

Question

I am stuck on the following problem that says:

How many complex numbers $z=x+iy$ are there such that $x+y=1$ and $e^{i(x^2+y^2)}=1.$ The options are as follows:
$1.0$
$2.$Non-zero but finitely many
$3.$Countably infinite
$4.$Uncountably infinite.

My Attempt: From $e^{i(x^2+y^2)}=1= e^{i(2n \pi)}$ which gives $x^2+y^2=2n \pi$ (where $n \in \mathbb N$) that indicates family of concentric circles with center at the origin. The required solution is the intersection of $x^2+y^2=2n \pi$ and the line $x+y=1$.But now I can not draw the conclusion.Am I going in the right direction? Can someone throw light on it.Thanks in advance for your time.

Answer 1

$x+y=1$ describes a line. Any sufficiently big circle around the origin intersects this lline in two points. So we must have countably many solutions.

Answer 2

Yes, but now consider how many times do the line and circles intersect. Looks like it would be either option 3 or 4 though one is more likely if you consider the domain of n. How many unique values of n exist?