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I asked the same question but with some fatal mistake that makes the question unanswerable - so I decided to delete it and start new.

Connecting from The set of numbers that when multiplied do not get decomposed into $sx+ty$ while the numbers themselves are of form $ax+byi$ (it's however an independent question.):

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).

Let us say call the real part of each number as $a_i$ and the integer coefficient of imaginary part as $b_i$. Let us say that $a_i = r_{i_1}r_{i_2}d$ and $b_i = r_{i_3}r_{i_4}d$ where $d$ is greatest common divisor of all $a_i$s and $b_i$s. $r_{i_1} = p_{i_1}^{e_{i_1}}$ where $p_{i_1}$ is the prime factor that only one(each) number's $a_i$ has, and $e_{i_1}$ is the corresponding exponent to the prime factor. (for readability, I will write: pi1, ei1)

$r_{i_3} = p_{i_3}^{e_{i_3}}$ where $p_{i_3}$ is the prime factor that only one number's $b_i$ has, and $e_{i_3}$ is the corresponding exponent to the prime factor. (pi3,ei3)

$r_{i_2}, r_{i_4}$ can be any non-zero integer.

Let us say that $t_1 = \text{product of all $r_{i_1}$'s}$ and $t_2 = \text{product of all $r_{i_2}$'s}$.

We would like to impose condition on these numbers so that out of all possible multiplication products of $n$ numbers (that is, this set includes multiplication that has duplicate number), only when $n$ different numbers are multiplied the product can be written as the form: $k_3t_1 + k_4d + k_5t_2$ where $k_3,k_4,k_5$ can be any non-zero integer.

Is this possible?

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So, you are assuming that for each $i$ there is a prime that divides $a_i$ and only $a_i$, and a prime that divides $b_i$ and only divides $b_i$? Also, $t_1$ and $t_2$ relate only to the $a_i$ and have nothing to do with the $b_i$? Also, with all this set-up, I'd expect a significant payoff; what good will it do you to have an answer to this problem? Where is it going? –  Gerry Myerson Jan 30 '13 at 4:01
    
yes, and I'm doing this because of economics dynamic issue. But I don't think this qualifies as needing economics tag. –  Siona Jan 30 '13 at 4:22
    
I'll give my 150 reputation points for a person who solves the problem. –  user27515 Jan 30 '13 at 6:08
    
@user27515, there is a way to offer a bounty on a question. See the faq. –  Gerry Myerson Jan 30 '13 at 6:25
    
Siona, tag doesn't worry me, would just be nice to understand the motivation. Maybe one could get away with less? –  Gerry Myerson Jan 30 '13 at 6:27

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