![Squared right triangle](https://upload.wikimedia.org/wikipedia/commons/3/3c/Squared_right_triangle.svg)
A Pythagorean prime is a
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
of the Pythagorean primes are exactly the odd prime numbers that are the sum of two squares; this characterization is
Fermat's theorem on sums of two squares
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime ''p'' can be expressed as:
:p = x^2 + y^2,
with ''x'' and ''y'' integers, if and only if
:p \equiv 1 \pmod.
The prime numbers for which this is true ar ...
.
Equivalently, by the
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
, they are the odd prime numbers
for which
is the length of the
hypotenuse
In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equa ...
of a
right triangle
A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
with integer legs, and they are also the prime numbers
for which
itself is the hypotenuse of a primitive
Pythagorean triangle
A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A primitive Pythagorean triple is ...
. For instance, the number 5 is a Pythagorean prime;
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.
Values and density
The first few Pythagorean primes are
By
Dirichlet's theorem on arithmetic progressions
In number theory, Dirichlet's theorem, also called the Dirichlet prime number theorem, states that for any two positive coprime integers ''a'' and ''d'', there are infinitely many primes of the form ''a'' + ''nd'', where ''n'' is als ...
, this sequence is infinite. More strongly, for each
, the numbers of Pythagorean and non-Pythagorean primes up to
are approximately equal. However, the number of Pythagorean primes up to
is frequently somewhat smaller than the number of non-Pythagorean primes; this phenomenon is known as For example, the only values of
up to 600000 for which there are more Pythagorean than non-Pythagorean odd primes less than or equal to n are 26861
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 number
A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...
s such as 21 that are and yet cannot be represented as sums of two squares.
Fermat's theorem on sums of two squares
In additive number theory, Fermat's theorem on sums of two squares states that an odd prime ''p'' can be expressed as:
:p = x^2 + y^2,
with ''x'' and ''y'' integers, if and only if
:p \equiv 1 \pmod.
The prime numbers for which this is true ar ...
states that the
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
s that can be represented as sums of two squares are exactly 2 and the odd primes congruent to The representation of each such number is unique, up to the ordering of the two squares.
By using the
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
, this representation can be interpreted geometrically: the Pythagorean primes are exactly the odd prime numbers
such that there exists a
right triangle
A right triangle (American English) or right-angled triangle (British), or more formally an orthogonal triangle, formerly called a rectangled triangle ( grc, ὀρθόσγωνία, lit=upright angle), is a triangle in which one angle is a right an ...
, with integer legs, whose
hypotenuse
In geometry, a hypotenuse is the longest side of a right-angled triangle, the side opposite the right angle. The length of the hypotenuse can be found using the Pythagorean theorem, which states that the square of the length of the hypotenuse equa ...
has They are also exactly the prime numbers
such that there exists a right triangle with integer sides whose hypotenuse has For, if the triangle with legs
and
has hypotenuse length
(with
), then the triangle with legs
and
has hypotenuse
Another way to understand this representation as a sum of two squares involves
Gaussian integer
In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with ordinary addition and multiplication of complex numbers, form an integral domain, usually written as \mathbf /ma ...
s, the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s whose real part and imaginary part are both The norm of a Gaussian integer
is the 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
Similarly, their squares can be factored in a different way than their
integer factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization.
When the numbers are suf ...
, 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
In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it has many formulations, but the most standard st ...
says that if
and
are distinct odd primes, at least one of which is Pythagorean, then
is a
quadratic residue if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicondi ...
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
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
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
![Paley13](https://upload.wikimedia.org/wikipedia/commons/0/05/Paley13.svg)
For every Pythagorean prime
, there exists a
Paley graph
In mathematics, Paley graphs are dense undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, ...
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
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Classes of prime numbers
Squares in number theory
{{Prime number classes, state=collapsed