Pythagorean triplet
That is, is a Pythagorean triplet if there exists a right triangle whose sides have lengths , , and , respectively. For example, is a Pythagorean triplet. Given one Pythagorean triplet , we can produce another by multiplying , , and by the same factor . It follows that there are countably many Pythagorean triplets.
Primitive Pythagorean triplets
A Pythagorean triplet is primitive if its elements are coprimes. All primitive Pythagorean triplets are given by
(1) |
where the seed numbers and are any two coprime positive integers, one odd and one even, such tht . If one presumes of the positive integers and only that , one obtains also many non-primitive triplets, but not e.g. . For getting all, one needs to multiply the right hand sides of (1) by an additional integer parametre .
Note 1. Among the primitive Pythagorean triples, the odd cathetus may attain all odd values except 1 (set e.g. ) and the even cathetus all values divisible by 4 (set ).
Note 2. In the primitive triples, the hypothenuses 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 integer as its real part, imaginary part and absolute value.
Note 4. The equations (1) imply that the sum of a cathetus and the hypotenuse is always a perfect square or a double perfect square.
Note 5. One can form the sequence (cf. Sloane’s http://www.research.att.com/ njas/sequences/?q=A100686&language=english&go=SearchA100686)
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 |