APythagorean prime is aprime number of theform${\displaystyle 4n+1}$. Pythagorean primes are exactly the odd prime numbers that are the sum of two squares; this characterization isFermat's theorem on sums of two squares.

Equivalently, by thePythagorean theorem, they are the odd prime numbers${\displaystyle p}$ for which${\displaystyle \sqrt{p}}$ is the length of thehypotenuse of aright triangle with integer legs, and they are also the prime numbers${\displaystyle p}$ for which${\displaystyle p}$ itself is the hypotenuse of a primitivePythagorean triangle. For instance, the number 5 is a Pythagorean prime;${\displaystyle \sqrt{5}}$ is the hypotenuse of a right triangle with legs 1 and 2, and 5 itself is the hypotenuse of a right triangle with legs 3 and 4.

The first few Pythagorean primes are

ByDirichlet's theorem on arithmetic progressions, this sequence is infinite. More strongly, for each${\displaystyle n}$, the numbers of Pythagorean and non-Pythagorean primes up to${\displaystyle n}$ are approximately equal. However, the number of Pythagorean primes up to${\displaystyle n}$ is frequently somewhat smaller than the number of non-Pythagorean primes; this phenomenon is known asChebyshev's bias.^[1] For example, the only values of${\displaystyle n}$ up to 600,000 for which there are more Pythagorean than non-Pythagorean odd primes less than or equal to n are 26861and 26862.^[2]

The sum of one odd square and one even square is congruent to 1 mod 4, but there existcomposite numbers such as 21 that are1 mod 4 and yet cannot be represented as sums of two squares.Fermat's theorem on sums of two squares states that theprime numbers that can be represented as sums of two squares are exactly 2 and the odd primes congruent to1 mod 4.^[3] The representation of each such number is unique, up to the ordering of the two squares.^[4]

By using thePythagorean theorem, this representation can be interpreted geometrically: the Pythagorean primes are exactly the odd prime numbers${\displaystyle p}$ such that there exists aright triangle, with integer legs, whosehypotenuse haslength${\displaystyle \sqrt{p}}$. They are also exactly the prime numbers${\displaystyle p}$ such that there exists a right triangle with integer sides whose hypotenuse haslength${\displaystyle p}$. 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 hypotenuselength${\displaystyle p}$.^[5]

Another way to understand this representation as a sum of two squares involvesGaussian integers, thecomplex numbers whose real part and imaginary part are bothintegers.^[6] The norm of a Gaussian integer${\displaystyle x+iy}$ is thenumber${\displaystyle x^2+y^2}$. 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 theirinteger factorization, asThe real and imaginary parts of the factors in these factorizations are the leg lengths of the right triangles having the given hypotenuses.

The law ofquadratic reciprocity says that if${\displaystyle p}$ and${\displaystyle q}$ are distinct odd primes, at least one of which is Pythagorean, then${\displaystyle p}$ is aquadratic residuemod${\displaystyle q}$if and only if${\displaystyle q}$ is a quadratic residuemod${\displaystyle p}$; by contrast, if neither${\displaystyle p}$ nor${\displaystyle q}$ is Pythagorean, then${\displaystyle p}$ is a quadratic residuemod${\displaystyle q}$ if and only if${\displaystyle q}$ isnot a quadratic residuemod${\displaystyle p}$.^[4]

In thefinite 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 residuemod${\displaystyle p}$. In contrast, this equation has no solution in the finite fields${\displaystyle \mathbb{Z}/p}$ where${\displaystyle p}$ is an odd prime but is notPythagorean.^[4]

For every Pythagorean prime${\displaystyle p}$, there exists aPaley graph with${\displaystyle p}$ vertices, representing the numbersmodulo${\displaystyle p}$, 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 quadraticresidue.^[7]

• Eaves, Laurence,"Pythagorean Primes: including 5, 13 and 137",Numberphile,Brady Haran, archived fromthe original on 2016-03-19, retrieved2013-04-02
• OEISsequence A007350 (Where prime race 4n-1 vs. 4n+1 changes leader)

• Classes of prime numbers
• Squares in number theory

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