Pythagorean prime: Difference between revisions
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==Representation as a sum of two squares== |
==Representation as a sum of two squares== |
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Every odd sum of two squares must be congruent to 1 mod 4, but there exist numbers such as 21 that are 1 mod 4 and yet cannot be represented as sums of two squares. |
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[[Fermat's theorem on sums of two squares]] states that the [[prime number]]s that can be represented as sums of two squares are exactly 2 and the odd primes congruent to 1 mod 4.<ref>{{citation|title=Why Beauty is Truth: A History of Symmetry|first=Ian|last=Stewart|authorlink=Ian Stewart (mathematician)|publisher=Basic Books|year=2008|isbn=9780465082377|page=264|url=http://books.google.com/books?id=6akF1v7Ds3MC&pg=PA264}}.</ref> The representation of each such number is unique, up to the ordering of the two squares, and the only odd numbers that have a unique representation as a sum of two squares are the Pythagorean primes.<ref>{{citation|title=Fundamentals of Number Theory|first=William Judson|last=LeVeque|authorlink=William J. LeVeque|publisher=Dover|year=1996|isbn=9780486689067|page=183|url=http://books.google.com/books?id=F6aJtNcwyw8C&pg=PA183}}.</ref> |
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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 {{sqrt|''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 {{sqrt|''p''}} (with ''x'' > ''y''), then the triangle with sides ''x''<sup>2</sup> − ''y''<sup>2</sup> and 2''xy'' has hypotenuse length ''p''. |
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 {{sqrt|''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 {{sqrt|''p''}} (with ''x'' > ''y''), then the triangle with sides ''x''<sup>2</sup> − ''y''<sup>2</sup> and 2''xy'' has hypotenuse length ''p''.<ref>{{citation|title=Elements of Number Theory|series=Undergraduate Texts in Mathematics|first=John|last=Stillwell|authorlink=John Stillwell|publisher=Springer|year=2003|isbn=9780387955872|page=112|url=http://books.google.com/books?id=LiAlZO2ntKAC&pg=PA112}}.</ref> |
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Another way to understand this representation as a sum of two squares involves [[Gaussian integer]]s, the [[complex number]]s whose real part and imaginary part are both integers. |
Another way to understand this representation as a sum of two squares involves [[Gaussian integer]]s, the [[complex number]]s whose real part and imaginary part are both integers. |
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[[File:Paley13.svg|thumb|The Paley graph with 13 vertices]] |
[[File:Paley13.svg|thumb|The Paley graph with 13 vertices]] |
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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 two numbers are subtracted to compute their difference, because of the property of Pythagorean primes that −1 is a quadratic residue. |
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 two numbers are subtracted to compute their difference, because of the property of Pythagorean primes that −1 is a quadratic residue. |
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==References== |
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{{reflist}} |
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==External links== |
==External links== |
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Revision as of 01:38, 15 November 2013
 | This article needs additional citations for verification. (November 2007) |
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
Representation as a sum of two squares
Every odd sum of two squares must be congruent to 1 mod 4, but there exist 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.[1] The representation of each such number is unique, up to the ordering of the two squares, and the only odd numbers that have a unique representation as a sum of two squares are the Pythagorean primes.[2]
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.[3]
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.
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 two numbers are subtracted to compute their difference, because of the property of Pythagorean primes that −1 is a quadratic residue.
References
- ^ Stewart, Ian (2008), Why Beauty is Truth: A History of Symmetry, Basic Books, p. 264, ISBN 9780465082377.
- ^ LeVeque, William Judson (1996), Fundamentals of Number Theory, Dover, p. 183, ISBN 9780486689067.
- ^ Stillwell, John (2003), Elements of Number Theory, Undergraduate Texts in Mathematics, Springer, p. 112, ISBN 9780387955872.
External links
- Eaves, Laurence. "Pythagorean Primes: including 5, 13 and 137". Numberphile. Brady Haran.