Difference between revisions of "Pythagorean triple"
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| − | A '''Pythagorean | + | A '''Pythagorean triple''' is a triple of [[positive integer]]s, <math>(a, b, c)</math> such that <math>a^2 + b^2 = c^2</math>. Pythagorean triples arise in [[geometry]] as the side-lengths of [[right triangle]]s. |
== Common Pythagorean Triples == | == Common Pythagorean Triples == | ||
| − | These are some common Pythagorean | + | These are some common Pythagorean triples: |
(3, 4, 5) | (3, 4, 5) | ||
| Line 37: | Line 37: | ||
== Primitive Pythagorean Triples == | == Primitive Pythagorean Triples == | ||
| − | A Pythagorean | + | A Pythagorean triple is called ''primitive'' if its three members have no common [[divisor]]s, so that they are [[relatively prime]]. All of the above triples are primitive. Integral [[multiple]]s of the above triples will also satisfy <math>a^2 + b^2 = c^2</math>, but they will not form primitive triples. For example, any three numbers in the form of <math>(3x, 4x, 5x)</math>, such as <math>(6, 8, 10)</math>, will also satisfy it. |
== See also == | == See also == | ||
* [[Pythagorean Theorem]] | * [[Pythagorean Theorem]] | ||
* [[Diophantine equation]] | * [[Diophantine equation]] | ||
Revision as of 01:56, 3 March 2007
A Pythagorean triple is a triple of positive integers, 
such that 
. Pythagorean triples arise in geometry as the side-lengths of right triangles.
Common Pythagorean Triples
These are some common Pythagorean triples:
(3, 4, 5)
(20, 21, 29)
(11, 60, 61)
(13, 84, 85)
(5, 12, 13)
(12, 35, 37)
(16, 63, 65)
(36, 77, 85)
(8, 15, 17)
(9, 40, 41)
(33, 56, 65)
(39, 80, 89)
(7, 24, 25)
(28, 45, 53)
(48, 55, 73)
(65, 72, 97)
Primitive Pythagorean Triples
A Pythagorean triple is called primitive if its three members have no common divisors, so that they are relatively prime. All of the above triples are primitive. Integral multiples of the above triples will also satisfy , but they will not form primitive triples. For example, any three numbers in the form of 
, such as 
, will also satisfy it.
