Pythagorean prime

Pythagorean prime

A Pythagorean prime is prime number of the form 4"n" + 1. These are exactly the primes that can be the hypotenuse of a Pythagorean triangle.

The first few Pythagorean primes are:5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, … OEIS|id=A002144.

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.

The law of quadratic reciprocity says that if "p" and "q" are 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". −1 is a quadratic residue mod "p" if and only if "p" is a Pythagorean prime (or 2).


Wikimedia Foundation. 2010.

Игры ⚽ Нужно решить контрольную?

Look at other dictionaries:

  • Pythagorean — means of or pertaining to the ancient Ionian mathematician, philosopher, and music theorist Pythagoras. See:Philosophy* Pythagoreanism an Egyptian influenced Neo Platonic philosophy named after the Greek philosopher and mathematician,… …   Wikipedia

  • Pythagorean triple — A Pythagorean triple consists of three positive integers a , b , and c , such that a 2 + b 2 = c 2. Such a triple is commonly written ( a , b , c ), and a well known example is (3, 4, 5). If ( a , b , c ) is a Pythagorean triple, then so is ( ka …   Wikipedia

  • Integer sequence prime — In mathematics, an integer sequence prime is a prime number found as a member of an integer sequence. For example, the 8th Delannoy number, 265729, is prime. A challenge in empirical mathematics is to identify large prime values in rapidly… …   Wikipedia

  • List of prime numbers — This is an incomplete list, which may never be able to satisfy particular standards for completeness. You can help by expanding it with reliably sourced entries. By Euclid s theorem, there are an infinite number of prime numbers. Subsets of the… …   Wikipedia

  • Mersenne prime — Named after Marin Mersenne Publication year 1536[1] Author of publication Regius, H. Number of known terms 47 Conjectured number of terms Infinite …   Wikipedia

  • Cuban prime — A cuban prime is a prime number that is a solution to one of two different specific equations involving third powers of x and y. The first of these equations is: and the first few cuban primes from this equation are (sequence A002407 in OEIS): 7 …   Wikipedia

  • List of mathematics articles (P) — NOTOC P P = NP problem P adic analysis P adic number P adic order P compact group P group P² irreducible P Laplacian P matrix P rep P value P vector P y method Pacific Journal of Mathematics Package merge algorithm Packed storage matrix Packing… …   Wikipedia

  • 257 (number) — 257 is the natural number between 256 and 258. It is also a prime number.Number|number = 257 range = 0 1000 cardinal = two hundred [and] fifty seven ordinal = th ordinal text = numeral = factorization = prime prime = divisor = roman = CCLVII… …   Wikipedia

  • 269 (number) — 269 is the natural number between 268 and 270. It is also a prime number.In mathematics269 is a regular prime, an Eisenstein prime with no imaginary part, a long prime, a Chen prime, a Pillai prime, a Pythagorean prime, a twin prime, a sexy prime …   Wikipedia

  • 313 (number) — Number|number = 313 range = 0 1000 cardinal = Three hundred thirteen ordinal = th ordinal text = numeral = factorization = prime prime = twin, palindromic divisor = roman = CCCXIII unicode = greek prefix = latin prefix = bin = 100111001 oct = duo …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”