2006 SMT/Algebra Problems/Problem 3
Problem
A Gaussian prime is a Gaussian integer (where
and
are integers) with no Guassian integer factors of smaller absolute value. Factor
into Gaussian primes with positive real parts.
is a symbol with the property that
.
Solution
Let . Therefore, we want to have
and
. Since
, we need
. First we try
. In this case,
, but this doesn't satisfy the second equality. Next we try
. First, we try
. In this case, we have
, so either
and
or
and
. However, neither of these satisfy the second equality. Next we try
. Again, either
and
or
and
. Checking, we find that
works. Therefore,
. Clearly, we cannot factor this any further.