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Pythagorean prime
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"Pythagorean number" redirects here. For the field invariant related to sums of squares, see Pythagoras number. For elements of extension fields containing square roots of sums of squares, see Pythagorean field.
A Pythagorean prime is a prime number of the form . 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.
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Equivalently, by the Pythagorean theorem, they are the odd prime numbers for which
is the length of the hypotenuse of a right triangle with integer legs, and they are also the prime numbers
for which
itself is the hypotenuse of a primitive Pythagorean triangle. For instance, the number 5 is a Pythagorean prime;
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.