Let $z=(x+yi)$
$ x \in \mathbb{Z} , y \in \mathbb{Z} $
I am searching for complex numbers which satisfy following equations:
$|\Re(z^2)| \equiv |\Re(z)| \; ( \bmod \; b^{\lfloor\log_{b}(|\Re(z)|)+1\rfloor} ) \wedge $
$ |\Im(z^2)| \equiv |\Im(z)| \; ( \bmod \; b^{\lfloor\log_{b}(|\Im(z)|)+1\rfloor} ) \; , b \in \mathbb{N} $
Which is:
$ |x^2 - y^2| \equiv |x| \; ( \bmod \; b^{\lfloor\log_{b}(|x|)+1\rfloor} ) \wedge$
$ |2xy| \equiv |y| \; ( \bmod \; b^{\lfloor\log_{b}(|y|)+1\rfloor} )$
The basic idea about these modulo equations is that $x^2−y^2$ ends with the digits of $x$ and $2xy$ ends with the digits of $y$, ignoring the signs. e.g. $−123$ ends with $23$.
$b$ is a base. I would be happy if it could be solved with $b=10$. But it would be even better if it could be solved for any base $b$.
An example of a complex number in base $b=10$:
(313 + 216i)² = 51313 + 135216i
So far, I have found 1447 numbers using a brute-force algorithm, but it is stuck and I wonder if there is more to find or if the amount of numbers was finite.
The smallest found number is $(8 + 4i)$, the biggest found number is $(959106445313 + 749939593216i)$. For some reason, extremely many numbers have $\Im(z)=60406784$.
Is there a rule or algorithm to find all possible $(x,y)$ tuples? Are there infinite numbers which satisfy these equations?