2006 SMT/Algebra Problems/Problem 3

Problem

A Gaussian prime is a Gaussian integer $z=a+bi$ (where $a$ and $b$ are integers) with no Guassian integer factors of smaller absolute value. Factor $-4+7i$ into Gaussian primes with positive real parts. $i$ is a symbol with the property that $i^2=-1$.

Solution

Let $-4+7i=(a+bi)(c+di)=(ac-bd)+(ad+bc)i$. Therefore, we want to have $ac=bd-4$ and $ad+bc=7$. Since $a, b, c, d>0$, we need $bd>4$. First we try $bd=5\implies b=1, d=5$. In this case, $ac=1\implies a=c=1$, but this doesn't satisfy the second equality. Next we try $bd=6$. First, we try $b=1, d=6$. In this case, we have $ac=2$, so either $a=2$ and $c=1$ or $a=1$ and $c=2$. However, neither of these satisfy the second equality. Next we try $b=2, d=3$. Again, either $a=2$ and $c=1$ or $a=1$ and $c=2$. Checking, we find that $(a,b,c,d)=(1,2,2,3)$ works. Therefore, $-4+7i=\boxed{(1+2i)(2+3i)}$. Clearly, we cannot factor this any further.

See Also

2006 SMT/Algebra Problems