proof of Pythagorean triples

If a,b, and c are positive integers such that

a2+b2=c2 (1)

then (a,b,c) is a Pythagorean tripleMathworldPlanetmath. If a,b, and c are relatively prime in pairs then (a,b,c) is a primitive Pythagorean triple. Clearly, if k divides any two of a,b, and c it divides all three. And if a2+b2=c2 then k2a2+k2b2=k2c2. That is, for a positive integer k, if (a,b,c) is a Pythagorean triple then so is (ka,kb,kc). Hence, to find all Pythagorean triples, it’s sufficient to find all primitive Pythagorean triples.

Let a,b, and c be relatively prime positive integers such that a2+b2=c2. Set


reduced to lowest terms, That is, gcd(m,n)=1. From the triangle inequality m>n. Then

mnb-a=c. (2)

Squaring both sides of (2) and multiplying through by n2 we get


which, after cancelling and rearranging terms, becomes

b(m2-n2)=a(2mn). (3)

There are two cases, either m and n are of opposite parity, or they or both odd. Since gcd(m,n)=1, they can not both be even.

Case 1. m and n of opposite parity, i.e., m±n(mod  2). So 2 divides b since m2-n2 is odd. From equation (2), n divides b. Since gcd(m,n)=1 then gcd(m,m2-n2)=1, therefore m also divides b. And since gcd(a,b)=1, b divides 2mn. Therefore b=2mn. Then

a=m2-n2,b=2mn,and from (2),c=mn 2mn-(m2-n2)=m2+n2. (4)

Case 2. m and n both odd, i.e., m±n(mod  2). So 2 divides m2-n2. Then by the same process as in the first case we have

a=m2-n22,b=mn,andc=m2+n22. (5)

The parametric equations in (4) and (5) appear to be different but they generate the same solutions. To show this, let

u=m+n2 and v=m-n2.

Then m=u+v, and n=u-v. Substituting those values for m and n into (5) we get

a=2uv,b=u2-v2,andc=u2+v2 (6)

where u>v, gcd(u,v)=1, and u and v are of opposite parity. Therefore (6), with a and b interchanged, is identical to (4). Thus since (m2-n2,2mn,m2+n2), as in (4), is a primitive Pythagorean triple, we can say that (a,b,c) is a primitive pythagorean triple if and only if there exists relatively prime, positive integers m and n, m>n, such that a=m2-n2,b=2mn, and c=m2+n2 .

Title proof of Pythagorean triplesPlanetmathPlanetmath
Canonical name ProofOfPythagoreanTriples
Date of creation 2013-03-22 14:28:05
Last modified on 2013-03-22 14:28:05
Owner fredlb (5992)
Last modified by fredlb (5992)
Numerical id 9
Author fredlb (5992)
Entry type Proof
Classification msc 11-00