Difference between revisions of "Pythagorean triple"
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== Primitive Pythagorean Triples == | == Primitive Pythagorean Triples == | ||
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. | 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. | ||
+ | |||
+ | === General Form of Primitive Pythagorean Triples === | ||
+ | |||
+ | '''Theorem.''' A triple of integers is a primitive Pythagorean triple if and only if it may be written in the form <math>(m^2-n^2, 2mn, m^2 + n^2)</math> or <math>(2mn, m^2-n^2, m^2+n^2)</math>, where <math>m > n</math> are relatively prime positive integers of different [[parity]]. | ||
+ | |||
+ | ==== Proof ==== | ||
+ | |||
+ | Let <math>(a,b,c)</math> be a primitive Pythagorean triple. If <math>a</math> and <math>b</math> both odd, then we must have | ||
+ | <cmath> c^2 \equiv a^2 + b^2 \equiv 1+1 \equiv 2 \pmod{4}, </cmath> | ||
+ | which is a contradiction, since 2 is not a [[quadratic residue | square]] [[modulus | mod]] 4. Hence at least one of <math>a</math> and <math>b</math>, say <math>b</math>, is even. Then <math>a</math> must be odd, since <math>a</math> and <math>b</math> must be relatively prime. It follows that <math>c</math> is odd as well. It follows that the numbers <math>s = (c+a)/2</math> and <math>d = (c-a)/2</math> are positive integers. These positive integers must be relatively prime, since any common divisor of <math>s</math> and <math>d</math> must divide both <math>s+d= c</math> and <math>s-d = a</math>. Since <math>a = s-d</math> and <math>c= s+d</math>, it follows that | ||
+ | <cmath> b = \sqrt{c^2 -a^2} = \sqrt{(s+d)^2 - (s-d)^2} = \sqrt{4sd} = 2\sqrt{sd} . </cmath> | ||
+ | Since <math>\sqrt{sd} = b/2</math> must be an integer and <math>s</math> and <math>d</math> are relatively prime, it follows that <math>s</math> and <math>d</math> are perfect squares. Hence we may denote <math>s= m^2</math> and <math>d = n^2</math> for integers <math>m</math> and <math>n</math>. Since <math>m^2 + n^2 = s+d = c</math> is odd, it follows that <math>m^2</math> and <math>n^2</math> must have different parity, so <math>m</math> and <math>n</math> have different parity. Finally, we observe that | ||
+ | <cmath> (m^2-n^2)^2 + (2mn)^2 = m^4 + 2m^2n^2 + n^4 = (m^2+n^2)^2, </cmath> | ||
+ | so any triple of the form specified in the theorem is a Pythagorean triple; it must furthermore be a primitive Pythagorean triple, since any common factor of <math>m^2 - n^2</math> and <math>m^2 + n^2</math> (both of which are odd integers, since <math>m</math> and <math>n</math> have different parity) must also be a factor of both <math>(m^2 + n^2) + (m^2-n^2) = 2m^2</math> and <math>(m^2+n^2) - (m^2 - n^2) = 2n^2</math>, which are integers with no common factor greater than 2. <math>\blacksquare</math> | ||
== See also == | == See also == | ||
* [[Pythagorean Theorem]] | * [[Pythagorean Theorem]] | ||
* [[Diophantine equation]] | * [[Diophantine equation]] |
Revision as of 21:53, 18 December 2007
A Pythagorean triple is a triple of positive integers, such that . Pythagorean triples arise in geometry as the side-lengths of right triangles.
Contents
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.
General Form of Primitive Pythagorean Triples
Theorem. A triple of integers is a primitive Pythagorean triple if and only if it may be written in the form or , where are relatively prime positive integers of different parity.
Proof
Let be a primitive Pythagorean triple. If and both odd, then we must have which is a contradiction, since 2 is not a square mod 4. Hence at least one of and , say , is even. Then must be odd, since and must be relatively prime. It follows that is odd as well. It follows that the numbers and are positive integers. These positive integers must be relatively prime, since any common divisor of and must divide both and . Since and , it follows that Since must be an integer and and are relatively prime, it follows that and are perfect squares. Hence we may denote and for integers and . Since is odd, it follows that and must have different parity, so and have different parity. Finally, we observe that so any triple of the form specified in the theorem is a Pythagorean triple; it must furthermore be a primitive Pythagorean triple, since any common factor of and (both of which are odd integers, since and have different parity) must also be a factor of both and , which are integers with no common factor greater than 2.