A Pythagorean prime is a prime leaving remainder 1 when divided by 4. By Fermat's theorem on sums of two squares, these are exactly the odd primes expressible as \(a^2 + b^2\): 5 = 1 + 4, 13 = 4 + 9, 17 = 1 + 16, 29 = 4 + 25.
The name comes from geometry: such a prime can be the hypotenuse of a right triangle with integer legs (5 in the 3–4–5 triangle, 13 in 5–12–13). Odd primes split evenly between the two classes mod 4 (Dirichlet), but in finite ranges the 4k+3 class almost always leads — the famous Chebyshev bias.