A Complex Situation

This happens when all the prime factors of $n$ are Gaussian primes as well, meaning that $n\equiv 3 \pmod{4}$ . The first such $n>50$ is $n=\boxed{57}=3\times 19$ . This Gaussian prime factorization is unique up to the units $\pm 1, \pm i$ .

To illustrate what happens when we have a prime factor 2 or $p\equiv 1 \pmod{4}$ , consider the factorizations $51= (3(4+i))(4-i)$ or $52=(26(1+i))(1-i)$ , for example. Recall that any $p\equiv 1 \pmod{4}$ can be written as $p=a^2+b^2=(a+bi)(a-bi)$ , by Fermat .