# 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. – Angina Seng Jul 26 '18 at 14:07
• And by no means is it useless! – Lubin Jul 26 '18 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. – Henrik supports the community Jul 26 '18 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. – Lubin Jul 26 '18 at 14:15