- Pythagorean prime
A Pythagorean prime is
prime number of the form 4"n" + 1. These are exactly the primes that can be thehypotenuse of aPythagorean triangle .The first few Pythagorean primes are:5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, … OEIS|id=A002144.
Fermat's theorem on sums of two squares states that these primes can be represented as sums of two squares uniquely (up to order), and that no other primes can be represented this way, aside from 2=12+12. Thus these primes (and 2) occur as norms ofGaussian integers , while other primes do not.The law of
quadratic reciprocity says that if "p" and "q" are odd primes, at least one of which is Pythagorean, then"p" is aquadratic residue mod "q"if and only if "q" is a quadratic residue mod "p"; by contrast, if neither "p" nor "q" is Pythagorean, then "p" is a quadratic residue mod "q" if and only if "q" is not a quadratic residue mod "p". −1 is a quadratic residue mod "p" if and only if "p" is a Pythagorean prime (or 2).
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