Pythagorean prime

A Pythagorean prime $p$ is a prime number of the form $4n+1$. The first few are 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, etc., listed in A002144 of Sloane’s OEIS. Because of its form, a Pythagorean prime is the sum of two squares, e.g., 29 = 25 + 4. In fact, with the exception of 2, these are the only primes that can be represented as the sum of two squares (thus, in Waring’s problem, all other primes require three or four squares).

Though Pythagorean primes are primes on the line of real integers, they are not Gaussian primes in the complex plane. Expressing a Pythagorean prime as $a^2+b^2$ (it doesn’t matter whether or viceversa) leads to the complex factorization by simple plugging in of the values thus: $p=(a+bi)(a-bi)$, where $i$ is the imaginary unit.