Pythagorean prime

Representation as a sum of two squares

The sum of one odd square and one even square is congruent to 1 mod 4, but there exist composite numbers such as 21 that are 1 mod 4 and yet cannot be represented as sums of two squares. Fermat's theorem on sums of two squares states that the prime numbers that can be represented as sums of two squares are exactly 2 and the odd primes congruent to 1 mod 4. The representation of each such number is unique, up to the ordering of the two squares.

By using the Pythagorean theorem, this representation can be interpreted geometrically: the Pythagorean primes are exactly the odd prime numbers ${\displaystyle p}$ such that there exists a right triangle, with integer legs, whose hypotenuse has length ${\displaystyle \sqrt{p}}$. They are also exactly the prime numbers ${\displaystyle p}$ such that there exists a right triangle with integer sides whose hypotenuse has length ${\displaystyle p}$. For, if the triangle with legs ${\displaystyle x}$ and ${\displaystyle y}$ has hypotenuse length ${\displaystyle \sqrt{p}}$ (with ${\displaystyle x>y}$), then the triangle with legs ${\displaystyle x^2-y^2}$ and ${\displaystyle 2xy}$ has hypotenuse length ${\displaystyle p}$.

Another way to understand this representation as a sum of two squares involves Gaussian integers, the complex numbers whose real part and imaginary part are both integers. The norm of a Gaussian integer ${\displaystyle x+iy}$ is the number ${\displaystyle x^2+y^2}$. Thus, the Pythagorean primes (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 $${\displaystyle p=(x+iy)(x-iy).}$$ Similarly, their squares can be factored in a different way than their integer factorization, as $${\displaystyle \begin{array}{cc}{p}^2& =(x+iy{)}^2(x-iy{)}^2\\ & =(x^2-y^2+2ixy)(x^2-y^2-2ixy).\end{array}}$$ 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 ${\displaystyle p}$ and ${\displaystyle q}$ are distinct odd primes, at least one of which is Pythagorean, then ${\displaystyle p}$ is a quadratic residue mod ${\displaystyle q}$ if and only if ${\displaystyle q}$ is a quadratic residue mod ${\displaystyle p}$; by contrast, if neither ${\displaystyle p}$ nor ${\displaystyle q}$ is Pythagorean, then ${\displaystyle p}$ is a quadratic residue mod ${\displaystyle q}$ if and only if ${\displaystyle q}$ is not a quadratic residue mod ${\displaystyle p}$.

In the finite field ${\displaystyle \mathbb{Z}/p}$ with ${\displaystyle p}$ a Pythagorean prime, the polynomial equation ${\displaystyle x^2=-1}$ has two solutions. This may be expressed by saying that ${\displaystyle -1}$ is a quadratic residue mod ${\displaystyle p}$. In contrast, this equation has no solution in the finite fields ${\displaystyle \mathbb{Z}/p}$ where ${\displaystyle p}$ is an odd prime but is not Pythagorean.

For every Pythagorean prime ${\displaystyle p}$, there exists a Paley graph with ${\displaystyle p}$ vertices, representing the numbers modulo ${\displaystyle p}$, 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 ${\displaystyle -1}$ is a quadratic residue.