Pythogerm Triples
Anmol Mehrotra
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
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