The Pythagorean prime is a prime of the form 4 n + 1.
Pythagorean primes can be represented as the sum of two squares (hence the name of the numbers - by analogy with the famous Pythagorean theorem .)
The first few primes of Pythagoras
- 5 , 13 , 17 , 29 , 37 , 41 , 53 , 61 , 73 , 89 , 97 , 101 , 109 , 113 , ... the sequence A002144 in OEIS .
The Fermat - Euler theorem states that these primes can be represented uniquely (up to an order) as the sum of two squares, and that no other primes can be represented in this way, except for 2 = 1 2 +1 2 . All these primes (including 2) are the norm of Gaussian integers , while other primes are not.
The quadratic reciprocity law states that if p and q are distinct odd primes, and at least one of them is Pythagorean, then p is a quadratic residue modulo q if and only if q is a quadratic residue modulo p ; and vice versa, if neither p nor q are Pythagorean, then p is a quadratic residue modulo q if and only if q is a square non-residue modulo p .
In the field Z / p with the Pythagorean prime p, the polynomial has two solutions.
