Pythagorean triplet

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

{

a

=

2

⁢

m

⁢

n

,

b

=

m

2

-

n

2

,

c

=

m

2

+

n

2

,

(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

+

i

⁢

n

)

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=Search

A100686)

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

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