(and 2) occur as norms of Gaussian integers, while other primes do not. Within the Gaussian integers, the Pythagorean primes are not considered to be prime numbers, because they can be factored as Similarly, their squares can be factored in a different way than their integer factorization, as The real and imaginary parts of the factors in these factorizations are the leg lengths of the right triangles having the given hypotenuses. Quadratic residues The law of quadratic reciprocity says that if and are distinct odd primes, at least one of which is Pythagorean, then is a quadratic residue if and only if is a quadratic residue by contrast, if neither nor is Pythagorean, then is a quadratic residue if and only if is not a quadratic residue In the finite field with a Pythagorean prime, the polynomial equation has two solutions. This may be expressed by saying that is a quadratic residue In contrast, this equation has no solution in the finite fields where is an odd prime but is not For every Pythagorean prime , there exists a Paley graph with vertices, representing the numbers with two numbers adjacent in the graph if and only if their difference is a quadratic residue. This definition produces the same adjacency relation regardless of the order in which the two numbers are subtracted to compute their difference, because of the property of Pythagorean primes that is a quadratic References External links Classes of prime numbers Squares in number theory