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