# Pythogerm Triples

**Topics:**Pythagorean theorem, Pythagorean triple, Euclidean algorithm

**Pages:**39 (8367 words)

**Published:**April 15, 2015

Pythagorean triples

Math Bonus

A

Pythagorean triple

consists of three positive

integers

a

,

b

, and

c

, such

2

2

2

that

a

+

b

=

c

. Such a triple is commonly written (

a

,

b

,

c

), and a

wellknown example is (3, 4, 5). If (

a

,

b

,

c

) is a Pythagorean triple, then so

is (

ka

,

kb

,

kc

) for any positive integer

k

. A

primitive Pythagorean triple

is

one in which

a

,

b

and

c

are

coprime

. A right triangle whose sides form a

Pythagorean triple is called a

Pythagorean triangle

.

The name is derived from the

Pythagorean theorem

, stating that every

right triangle

has side lengths satisfying the formula

a2

+

b2

=

c2

; thus,

Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with noninteger sides do not form Pythagorean triples. For instance, the

triangle

with sides

a

=

b

= 1 and

c

=

√2 is right, but (1, 1, √2) is not a Pythagorean triple because √2 is not an integer. Moreover, 1 and √2 do not have an integer common multiple because √2 is

irrational

.

Examples[

edit

]

A

scatter plot

of the legs (

a

,

b

) of the Pythagorean triples with

c

less

than 6000. Negative values are included to illustrate the parabolic patterns in the plot more clearly.

Contents

There are 16 primitive Pythagorean triples with

c

≤ 100:

(3, 4, 5)

(5, 12,

(8, 15,

(7, 24,

13)

17)

25)

(20, 21,

(12, 35,

(9, 40,

(28, 45,

29)

37)

41)

53)

(11, 60,

(16, 63,

(33, 56,

(48, 55,

61)

65)

65)

73)

(13, 84,

(36, 77,

(39, 80,

(65, 72,

85)

85)

89)

97)

Note, for example, that (6, 8, 10) is

not

a primitive Pythagorean triple, as it

is a multiple of (3, 4, 5). Each one of these lowc points forms one of the more easily recognizable radiating lines in the scatter plot. Additionally these are all the primitive Pythagorean triples with 100 ≤

300:

(20, 99,

(60, 91,

(15, 112,

(44, 117,

101)

109)

113)

125)

(88, 105,

(17, 144,

(24, 143,

(51, 140,

137)

145)

145)

149)

(85, 132,

(119, 120,

(52, 165,

(19, 180,

157)

169)

173)

181)

(57, 176,

(104, 153,

(95, 168,

(28, 195,

185)

185)

193)

197)

(84, 187,

(133, 156,

(21, 220,

(140, 171,

205)

205)

221)

221)

(60, 221,

(105, 208,

(120, 209,

(32, 255,

229)

233)

241)

257)

(23, 264,

(96, 247,

(69, 260,

(115, 252,

265)

265)

269)

277)

(160, 231,

281)

(161, 240,

289)

(68, 285,

293)

Generating a triple[

edit

]

Main article:

Formulas for generating Pythagorean triples

The primitive Pythagorean triples. The odd leg

a

is plotted on the

horizontal axis, the even leg

b

on the vertical. The curvilinear grid is

composed of curves of constant

m

n

and of constant

m

+

n

in Euclid's

formula.

A plot of triples generated by Euclid's formula map out part of the z2

=

x2

+

2

y

cone. A constant

m

or

n

traces out part of a

parabola

on the cone.

[1]

Euclid's formula

is a fundamental formula for generating Pythagorean triples given an arbitrary pair of positive integers m

and

n

with

m

>

n

. The

formula states that the integers

form a Pythagorean triple. The triple generated by

Euclid

's formula is

primitive if and only if

m

and

n

are

coprime

and

m

−

n

is odd. If both

m

and

n

are odd, then

a

,

b

, and

c

will be even, and so the triple will not be

primitive; however, dividing

a

,

b

, and

c...

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