## You are here

HomePythagorean triplet

## Primary tabs

# 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.

## Mathematics Subject Classification

11-00*no label found*01A20

*no label found*11A05

*no label found*11-03

*no label found*11-01

*no label found*51M05

*no label found*51M04

*no label found*51-03

*no label found*51-01

*no label found*01-01

*no label found*55-00

*no label found*55-01

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections

## Attached Articles

proof of Pythagorean triples by fredlb

first primitive Pythagorean triplets by pahio

proof of Pythagorean triples by rm50

rational sine and cosine by pahio

derivation of Pythagorean triples by pahio

geometric proof of Pythagorean triplet by rm50

rational points on one dimensional sphere by joking

linear formulas for Pythagorean triples by pahio