Pythagorean prime: Difference between revisions
Oops, it's the number itself (and not its square) that is the sum of two squares. Undo change for now. |
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==Representation as a sum of two squares== |
==Representation as a sum of two squares== |
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[[Fermat's theorem on sums of two squares]] states that these primes can be represented as sums of two squares uniquely (up to order), and that no other primes can be represented this way, aside from 2=1<sup>2</sup>+1<sup>2</sup>. Thus these primes (and 2) occur as norms of [[Gaussian integers]], while other primes do not. |
[[Fermat's theorem on sums of two squares]] states that these primes can be represented as sums of two squares uniquely (up to order), and that no other primes can be represented this way, aside from 2=1<sup>2</sup>+1<sup>2</sup>. Thus these primes (and 2) occur as norms of [[Gaussian integers]], while other primes do not. |
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By using the [[Pythagorean theorem]], this representation can be interpreted geometrically: the Pythagorean primes are exactly the 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''}}, then the triangle with sides ''x''<sup>2</sup> − ''y''<sup>2</sup> and 2''xy'' has hypotenuse length ''p''. |
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==Quadratic residues== |
==Quadratic residues== |
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Revision as of 21:45, 14 November 2013
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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 (and from this derives the name in reference to the famous Pythagorean theorem.)
The sequence of Pythagorean primes
The first few Pythagorean primes are
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 order), and that no other primes can be represented this way, aside from 2=12+12. Thus these primes (and 2) occur as norms of Gaussian integers, while other primes do not.
By using the Pythagorean theorem, this representation can be interpreted geometrically: the Pythagorean primes are exactly the 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, then the triangle with sides x2 − y2 and 2xy has hypotenuse length p.
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 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.
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