Pythagorean triplet

A Pythagorean triplet is a set {a,b,c} of three positive integers such that


That is, {a,b,c} is a Pythagorean triplet if there exists a right triangleMathworldPlanetmath whose sides have lengths a, b, and c, respectively. For example, {3,4,5} is a Pythagorean triplet. Given one Pythagorean triplet {a,b,c}, we can produce another by multiplying a, b, and c by the same factor k. It follows that there are countably many Pythagorean triplets.

Primitive Pythagorean triplets

A Pythagorean triplet is primitivePlanetmathPlanetmath if its elements are coprimesMathworldPlanetmath. All primitive Pythagorean triplets are given by

{a=2mn,b=m2-n2,c=m2+n2, (1)

where the seed numbers m and n are any two coprime positive integers, one odd and one even, such tht m>n.  If one presumes of the positive integers m and n only that  m>n, one obtains also many non-primitive triplets, but not e.g. (6, 8, 10).  For getting all, one needs to multiply the right hand sides of (1) by an additional integer parametre q.

Note 1.  Among the primitive Pythagorean triples, the odd cathetusMathworldPlanetmath a may attain all odd values except 1 (set e.g.  m:=n+1) and the even cathetus b all values divisible by 4 (set  n:=1).

Note 2.  In the primitive triples, the hypothenuses c are always odd.  All possible Pythagorean hypotenuses are contraharmonic means of two different integers (and conversely).

Note 3.  N.B. that any triplet (1) is obtained from the square of a Gaussian integerMathworldPlanetmath (m+in)2 as its real partMathworldPlanetmath, imaginary part and absolute valueMathworldPlanetmathPlanetmathPlanetmath.

Note 4.  The equations (1) imply that the sum of a cathetus and the hypotenuse is always a perfect squareMathworldPlanetmath or a double perfect square.

Note 5.  One can form the sequence (cf. Sloane’s njas/sequences/?q=A100686&language=english&go=SearchA100686)

1, 2, 3, 4, 7, 24, 527, 336, 164833, 354144,

taking first the seed numbers 1 and 2 which give the legs 3 and 4, taking these as new seed numbers which give the legs 7 and 24, and so on.

Title Pythagorean triplet
Canonical name PythagoreanTriplet
Date of creation 2013-03-22 11:43:48
Last modified on 2013-03-22 11:43:48
Owner drini (3)
Last modified by drini (3)
Numerical id 23
Author drini (3)
Entry type Definition
Classification msc 11-00
Classification msc 01A20
Classification msc 11A05
Classification msc 11-03
Classification msc 11-01
Classification msc 51M05
Classification msc 51M04
Classification msc 51-03
Classification msc 51-01
Classification msc 01-01
Classification msc 55-00
Classification msc 55-01
Synonym Pythagorean triple
Related topic PythagorasTheorem
Related topic IncircleRadiusDeterminedByPythagoreanTriple
Related topic ContraharmonicMeansAndPythagoreanHypotenuses
Related topic PythagoreanHypotenusesAsContraharmonicMeans
Defines seed number
Defines primitive Pythagorean triple
Defines primitive Pythagorean triplet