# Pythagorean triplet

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

 $a^{2}+b^{2}=c^{2}.$

That is, $\{a,b,c\}$ is a Pythagorean triplet if there exists a right triangle 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 primitive if its elements are coprimes. All primitive Pythagorean triplets are given by

 $\displaystyle\begin{cases}a=2mn,\\ b=m^{2}\!-\!n^{2},\\ c=m^{2}\!+\!n^{2},\end{cases}$ (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 cathetus $a$ 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 integer $(m\!+\!in)^{2}$ 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)

 $1,\,2,\,3,\,4,\,7,\,24,\,527,\,336,\,164833,\,354144,\,...$

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