Vipul: /* Classification */

2010-08-13T21:47:18Z

Classification

← Older revision		Revision as of 21:47, 13 August 2010
Line 9:		Line 9:
	==Classification==		==Classification==

	The classification of Pythagorean triples (up to the interchange symmetry of the first two elements) follows immediately from the [[classification of primitive Pythagorean triples]]. The set of Pythagorean triples is in bijection with the set of triples <math>(d,u,v)</math> of positive integers with <math>u < v</math>, with the bijection given by:		The classification of Pythagorean triples (up to the interchange symmetry of the first two elements) follows immediately from the [[classification of primitive Pythagorean triples]]. The set of Pythagorean triples is in bijection with the set of triples <math>(d,u,v)</math> of positive integers with <math>u < v</math>, <math>u,v</math> coprime, and one of <math>u</math> and <math>v</math> odd, with the bijection given by:

	<math>\! (d,u,v) \mapsto (d(v^2 - u^2), 2duv, d(u^2 + v^2))</math>		<math>\! (d,u,v) \mapsto (d(v^2 - u^2), 2duv, d(u^2 + v^2))</math>

Vipul: Created page with "==Definition== A '''Pythagorean triple''' is a triple $(a,b,c)$ of positive integers such that $a^2 + b^2 = c^2$ . Typically, the term ''Pythagorean triple..."

2010-08-13T21:30:45Z

Created page with "==Definition== A '''Pythagorean triple''' is a triple <math>(a,b,c)</math> of positive integers such that <math>a^2 + b^2 = c^2</math>. Typically, the term ''Pythagorean triple..."

New page

==Definition==

A '''Pythagorean triple''' is a triple <math>(a,b,c)</math> of positive integers such that <math>a^2 + b^2 = c^2</math>.

Typically, the term ''Pythagorean triple'' is used to refer to the triple up to the interchange symmetry of <math>a</math> and <math>b</math>. In other words, the triples <math>(a,b,c)</math> and <math>(b,a,c)</math> are considered equivalent.

We are typically interested in the study of [[primitive Pythagorean triple]]s, and often, the term ''Pythagorean triple'' is used for primitive Pythagorean triple. A primitive Pythagorean triple is a Pythagorean triple where the three elements are pairwise relatively prime, or equivalent, their overall gcd is <math>1</math>.

==Classification==

The classification of Pythagorean triples (up to the interchange symmetry of the first two elements) follows immediately from the [[classification of primitive Pythagorean triples]]. The set of Pythagorean triples is in bijection with the set of triples <math>(d,u,v)</math> of positive integers with <math>u < v</math>, with the bijection given by:

<math>\! (d,u,v) \mapsto (d(v^2 - u^2), 2duv, d(u^2 + v^2))</math>

Pythagorean triple - Revision history

Vipul: /* Classification */

Vipul: Created page with "==Definition== A '''Pythagorean triple''' is a triple (a,b,c) of positive integers such that a^2 + b^2 = c^2. Typically, the term ''Pythagorean triple..."

Vipul: Created page with "==Definition== A '''Pythagorean triple''' is a triple $(a,b,c)$ of positive integers such that $a^2 + b^2 = c^2$ . Typically, the term ''Pythagorean triple..."