Suppose that there are $n$ numbers that are in the following format: $a+bi$. Each number has different combination of $a$ and $b$. $a,b$ must be non-zero integers.
Suppose that we impose the following rule: $a^2-b^2 = k_1x$ and $2ab = k_2y$.
$k_1$ and $k_2$ are free non-zero integers - by free, I mean that they can be different for different numbers. However, $x$ and $y$ are fixed (set).
We want to set the number so that when all of these $n$ numbers are multiplied, the integer part of the multiplication result cannot equal to the form of $sx+ty$ where $s$ and $t$ can be any non-zero integers.
The question is,
is this possible? If so, would this be possible for any cardinality of the set of numbers less than infinite?