# Factorising complex numbers

4 years ago, when I learned about factorising and complex numbers, me and my friend worked on factorising complex numbers.

For example, $4+2i= 3-(-1)+2i = 3-i^2+2i = -(i^2-2i-3) = -(i-3)(i+1)$

The goal was to represent $a+bi$ with product of same form. where $a$ and $b$ are integer.

Another example is, $8+i= 8i^4+i=i(8i^3+1)=i(2i+1)(4i^2-2i+1)=i(2i+1)(-2i-3)=-i(2i+1)(2i+3)$

I showed my teacher, and she said it's useless.

Now I think of it, I don't know why I did this and it looks like same thing just in different form.

Is there any research already done on this or can there be any use of it?

Apparently,

$(n+2)+ni=-(i-(n+1))(i+1)$

$m^3+n^3i=-i(mi+n)(mni+(m^2-n^2))$

• Yes, there's a lot of work done on this. Look up Gaussian Integers. Jul 26, 2018 at 14:07
• And by no means is it useless! Jul 26, 2018 at 14:11
• There's research already done, but that does not mean there can't be any use of the work you do on this, when you've read up on the existing work. Jul 26, 2018 at 14:13
• I was thinking @Henrik, that knowing how to factor Gaussian integers is useful in what might look like other parts of mathematics. Jul 26, 2018 at 14:15