A Pythagorean prime is a prime number of the form ${\displaystyle 4n+1}$. Pythagorean primes are exactly the odd prime numbers that are the sum of two squares; this characterization is Fermat's theorem on sums of two squares.

Equivalently, by the Pythagorean theorem, they are the odd prime numbers ${\displaystyle p}$ for which ${\displaystyle {\sqrt {p}}}$ is the length of the hypotenuse of a right triangle with integer legs, and they are also the prime numbers ${\displaystyle p}$ for which ${\displaystyle p}$ itself is the hypotenuse of a primitive Pythagorean triangle. For instance, the number 5 is a Pythagorean prime; ${\displaystyle {\sqrt {5}}}$ is the hypotenuse of a right triangle with legs 1 and 2, and 5 itself is the hypotenuse of a right triangle with legs 3 and 4.