# integers

• Aug 8th 2009, 05:52 AM
geo2
integers
someone to help me with this

factorise 5550 fully in
1) the integers
2) the Gaussian integers?????????
• Aug 8th 2009, 06:56 AM
Opalg
Quote:

Originally Posted by geo2
someone to help me with this

factorise 5550 fully in
1) the integers
2) the Gaussian integers?????????

I assume you can do 1). (Wink)

For 2), you need to know that if p is an integer prime of the form 4k+3 then it is also a Gaussian prime. If it is of the form 4k+1 then it is always possible to express it as a sum of two squares, $p=a^2+b^2$. Then p factorises in the Gaussian integers as $p=(a+ib)(a-ib)$, and those factors are both Gaussian primes. Finally, 2 factorises in the same way, $2=(1+i)(1-i)$.

So for example 11 is a Gaussian prime, but $13 = 3^2+2^2 = (3+2i)(3-2i)$.