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Pythagorean primes

The Pythagorean primes are primes which are the sum of two squares, implying that among the Gaussian integers, these numbers are Gaussian composites

${\displaystyle p=a^{2}+b^{2}=(a-bi)(a+bi),\quad a,b\in \mathbb {N} ^{+}.\,}$

The even prime ${\displaystyle 2=1^{2}+1^{2}}$ and the odd primes of form ${\displaystyle 4n+1}$ are Pythagorean (sum of two squares), while the odd primes of form ${\displaystyle 4n-1}$ are not Pythagorean (not the sum of two squares).

A002144 Pythagorean odd primes: primes of form ${\displaystyle \scriptstyle 4n+1,\,n\,\geq \,1\,}$.

{5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, ...}

Non-Pythagorean primes

The non-Pythagorean primes are primes which are NOT the sum of two squares, implying that among the Gaussian integers, these numbers are Gaussian primes.

A002145 Non-Pythagorean primes: primes of form ${\displaystyle \scriptstyle 4n-1,\,n\,\geq \,1\,}$.

{3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 271, 283, 307, 311, 331, 347, 359, 367, 379, 383, 419, 431, ...}