Math 2.0

Number theory Theorem

Classification of Pythagorean triples

Builds on Pythagorean tripleCoprime integers

Statement

Let mm and nn be coprime positive integers with m>nm > n and m−nm - n odd. Then (m2−n2, 2mn, m2+n2)(m^2 - n^2, \, 2mn, \, m^2 + n^2) is a primitive Pythagorean triple. Conversely, each primitive Pythagorean triple (a,b,c)(a, b, c) with bb even arises in this way from exactly one such pair (m,n)(m, n). Each Pythagorean triple is (ka,kb,kc)(ka, kb, kc) for a positive integer kk and a primitive triple (a,b,c)(a, b, c), up to the order of the first two entries.

Remarks

The formula is called Euclid's formula. For example, (m,n)=(2,1)(m, n) = (2, 1) gives (3,4,5)(3, 4, 5), and (m,n)=(3,2)(m, n) = (3, 2) gives (5,12,13)(5, 12, 13). One proof writes b2=(c−a)(c+a)b^2 = (c - a)(c + a) and uses unique factorization. Another uses the rational points on the unit circle.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Divisibility, definition, no evidence levelDivisibilitydefinitionGreatest common divisor, definition, no evidence levelGreatest common divis…definitionCoprime integers, definition, no evidence levelCoprime integersdefinitionPythagorean triple, definition, no evidence levelPythagorean tripledefinitionClassification of Pythagorean triples, statement, no evidence levelClassification of Pyt…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Divisibility definition
  2. Greatest common divisor definition
  3. Coprime integers definition
  4. Pythagorean triple definition
  5. Classification of Pythagorean triples statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Pythagorean_triple, Generating a triple
About this page
Tag0KF3J2KAER
URIm2://0KF3J2KAER
Aliaseseuclid-formula
Kindstatement, theorem
WritingW0 stub
MSC11D09
StatusThis copy of the site has no build data. Run m2 status 0KF3J2KAER.
UsesPythagorean triple, Coprime integers

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