Problem - 2847
Find all prime numbers $p$ that can be written $p = x^4 + 4y^4$, where $x, y$ are positive integers.
The answer is $p = \boxed{5}$.
By Sophie Germain's Identity (see # 3863) we have $$x^4 + 4y^4 = (x^2 + 2y^2 + 2xy)(x^2 + 2y^2 - 2xy) = [(x + y)^2 + y^2][(x - y)^2 + y^2]$$
This can be a prime if one of them equals $1$ which leads to the conclusion that $x = y = 1$.