# 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

$\{\begin{array}{cc}a=2mn,\hfill & \\ b={m}^{2}-{n}^{2},\hfill & \\ c={m}^{2}+{n}^{2},\hfill & \end{array}$ | (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. $(6,\mathrm{\hspace{0.17em}8},\mathrm{\hspace{0.17em}10})$. 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)

$$1,\mathrm{\hspace{0.17em}2},\mathrm{\hspace{0.17em}3},\mathrm{\hspace{0.17em}4},\mathrm{\hspace{0.17em}7},\mathrm{\hspace{0.17em}24},\mathrm{\hspace{0.17em}527},\mathrm{\hspace{0.17em}336},\mathrm{\hspace{0.17em}164833},\mathrm{\hspace{0.17em}354144},\mathrm{\dots}$$ |

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 |