linear formulas for Pythagorean triples

It is easy to see that the equation

a2+b2=c2 (1)

of the Pythagorean theoremMathworldPlanetmathPlanetmath ( is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath ( with

(a+b-c)2= 2(c-a)(c-b). (2)

When  (a,b,c)  is a Pythagorean tripleMathworldPlanetmath, i.e. a, b, c are positive integers, a+b-c must be an even positive integer which we denote by 2r.  We get from (2) the equation

(c-a)(c-b)= 2r2,

whose factors ( on the left hand side we denote by t and s.  Thus we have the linear equation system

{a+b-c= 2r,c-a=t,c-b=s.

Its solution is

{a= 2r+s,b= 2r+t,c= 2r+s+t. (3)

Here, r is an arbitrary positive integer, s and t are two positive integers whose product is 2r2.  It’s clear that then (3) produces all Pythagorean triples.


  • 1 Egon Scheffold: “Ein Bild der pythagoreischen Zahlentripel”.  – Elemente der Mathematik 50 (1995).
Title linear formulasMathworldPlanetmathPlanetmath for Pythagorean triples
Canonical name LinearFormulasForPythagoreanTriples
Date of creation 2014-12-22 21:59:51
Last modified on 2014-12-22 21:59:51
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 13
Author pahio (2872)
Entry type Result
Classification msc 11-00
Related topic DerivationOfPythagoreanTriples
Related topic ContraharmonicMeansAndPythagoreanHypotenuses
Related topic DeterminingIntegerContraharmonicMeans