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.