Pythagorean triple

From Exampleproblems

A Pythagorean triple consists of three positive integers a, b, and c, such that a² + b² = c². Such a triple is commonly written (a, b, c), and a well-known 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.

The name is derived from the Pythagorean theorem, of which every Pythagorean triple is a solution. The converse is not true. 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 don't have an integer common multiple because √2 is irrational.

There are 16 primitive Pythagorean triples with c ≤ 100:

(3, 4, 5)	(20, 21, 29)	(11, 60, 61)	(13, 84, 85)
(5, 12, 13)	(12, 35, 37)	(16, 63, 65)	(36, 77, 85)
(8, 15, 17)	(9, 40, 41)	(33, 56, 65)	(39, 80, 89)
(7, 24, 25)	(28, 45, 53)	(48, 55, 73)	(65, 72, 97)

Generating Pythagorean triples

An effective way to generate Pythagorean triples is based on the observation that if m and n are two positive integers with m > n, then

a = m² − n²,

b = 2mn,

c = m² + n²

is a Pythagorean triple. It is primitive if and only if m and n are coprime and one of them is even (if both n and m are odd, then a, b, and c will be even, and so the Pythagorean triple will not be primitive). Not every Pythagorean triple can be generated in this way, but every primitive triple (possibly after exchanging a and b) arises in this fashion from a unique pair of coprime numbers m > n. This shows that there are infinitely many primitive Pythagorean triples.

See also

• Heronian triangle
• Platonic sequence

External links

• http://mathworld.wolfram.com/PythagoreanTriple.html has an extensive discussion of Pythagorean triples.
• http://www.math.clemson.edu/~rsimms/neat/math/pyth/ provides a Javascript calculator for the (m² − n², 2mn, m² + n²) formula, and shows how to derive the formula.
• http://www.faust.fr.bw.schule.de/mhb/pythagen.htm a JavaScript calculator which illustrates the 3-fold tree structure of the set of all primitive Pythagorean triples.
• Pythagorean Triples Where the formula comes from: interactive Java illustration.
• The Trinary Tree(s) underlying Primitive Pythagorean Triples by H. Andres Lönnemo
• Fermat's Last Theorem Blog Covers topics in the history of Fermat's Last Theorem from Pythagorean triples to Wiles' proof.bg:Питагоров триъгълник

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