: Mathématiques: Pythagorean prime

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Pythagorean prime

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.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Pythagorean_prime)

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