# Finding all $z$ which satisfy $|\Re(z^2)| \equiv |\Re(z)|$ and $|\Im(z^2)| \equiv |\Im(z)|$

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?

• ??? What is $b$? – Robert Israel Jul 15 '20 at 15:12
• What is the absolute value in the modulo world? – Christian Blatter Jul 15 '20 at 15:16
• @ChristianBlatter My guess is that by $|s|\equiv |t| \mod m$ the OP could mean: $s\equiv t \mod m$ or $s \equiv -t \mod m$. – Robert Israel Jul 15 '20 at 15:48
• Sorry, I forgot to define $b$. It 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 $b$. The basic idea about these modulo equations is that $2xy$ ends with the digits of $y$, and $x^2 - y^2$ ends with the digits if $x$, ignoring the signs. e.g. $-123$ ends with $23$. – Daniel Marschall Jul 15 '20 at 17:15