Wait! What about:
$$ 5 = -i \cdot (1 + 2i) \cdot (2 + i)$$
This is an example of a factorization in the quadratic field of Gaussian Integers. It is therefore not enough to say that a number is prime. Primality is relative in relation to the number field.
The mathematician Lamé thought to have cracked Fermat's Last Theorem in 1847. Needless to say that his proof contained an error. He overlooked the fact that prime factorization was not unique in a number system he used in his proof.
In Mathematica factorization is done with:
factorization using Gaussian integers is done with:
FactorInteger[5, GaussianIntegers -> True].