Pythagorean prime

"Pythagorean number" redirects here. For the field invariant related to sums of squares, see Pythagoras number.

The Pythagorean prime 5 and its square root are both hypotenuses of integer-sided right triangles.

A Pythagorean prime is a prime number of the form 4n + 1. Pythagorean primes are exactly the odd prime numbers that are the sum of two squares.

Equivalently, by the Pythagorean theorem, they are the odd prime numbers p for which √p is the length of the hypotenuse of a right triangle with integer sides, and they are also the prime numbers p for which p itself is the hypotenuse of a Pythagorean triangle. For instance, the number 5 is a Pythagorean prime; √5 is the hypotenuse of a right triangle with sides 1 and 2, and 5 itself is the hypotenuse of a right triangle with sides 3 and 4.

The sequence of Pythagorean primes

The first few Pythagorean primes are

5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, … (sequence A002144 in the OEIS).

Representation as a sum of two squares

Fermat's theorem on sums of two squares states that these primes can be represented as sums of two squares uniquely (up to the ordering of the two squares), and that no other primes can be represented this way, aside from 2 = 1² + 1².

By using the Pythagorean theorem, this representation can be interpreted geometrically: the Pythagorean primes are exactly the odd prime numbers p such that there exists a right triangle, with integer sides, whose hypotenuse has length √p. They are also exactly the prime numbers p such that there exists a right triangle with integer sides whose hypotenuse has length p. For, if the triangle with sides x and y has hypotenuse length √p (with x > y), then the triangle with sides x² − y² and 2xy has hypotenuse length 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 x + yi is the number x² + y². 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

p = (x + yi)(x − yi).

Similarly, their squares can be factored in a different way than their integer factorization, as

p² = (x + yi)²(x − yi)² = (x² − y² + 2xyi)(x² − y² − 2xyi).

The real and imaginary parts of the factors in these factorizations are the side lengths of the right triangles having the given hypotenuses.

Quadratic residues

The law of quadratic reciprocity says that if p and q are distinct odd primes, at least one of which is Pythagorean, then p is a quadratic 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.

In the finite field Z/p with p a Pythagorean prime, the polynomial equation x² =&nsbp;−1 has two solutions. This may be expressed by saying that −1 is a quadratic residue mod p. In contrast, this equation has no solution in the finite fields Z/p where p is a prime but is not Pythagorean.

The Paley graph with 13 vertices

For every Pythagorean prime p, there exists a Paley graph with p vertices, representing the numbers modulo 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 numbers are subtracted to compute their difference, because of the property of Pythagorean primes that −1 is a quadratic residue.

External links

• Eaves, Laurence. "Pythagorean Primes: including 5, 13 and 137". Numberphile. Brady Haran.