Primitive Pythagorean Triples
D Joyce, Clark University
March 2006, March 2010
This is a table of primitive Pythagorean triples. A Pythagorean triple (x, y, z) is a triple of positive
integers such that x2+y2=z2. It is primitive if the greatest common divisor of x,y, and zis 1. For any
primitive Pythagorean triple, either xor yis even, but not both, so we may always choose xto be o dd. When
that is done, then every primitive Pythagorean triple (x, y, z) is of the form
(x, y, z) = ab, a2−b2
2,a2+b2
2
where the parameters aand bare relatively prime odd numbers, 1 ≤b<a.
This parameterization is the one given by Euclid in Proposition 29 of Book X of his Elements. See
http://aleph0.clarku.edu/~djoyce/java/elements/bookX/prop29.html
Here’s a short explanation why these are all the primitive Pythagorean triples. First, we can move the y2
to the other side of the equation x2+y2=z2and rewrite it as x2=z2−y2so that we can factor the the
new right side as (z+y)(z−y). Next, we can divide by x2. Then we have 1 = z
x+y
xz
x−y
x. Thus, the
two terms on the right are reciprocals of each other. Let the first one be a/b in lowest terms, so the second
is b/a. Thus
z
x+y
x=a
b
z
x−y
x=b
a
Solving those two equations for z/x and y/x we find
z
x=1
2a
b+b
a=a2+b2
2ab
y
x=1
2a
b−b
a=a2−b2
2ab
Note that b<a. Since x,y, and zhave no common factor, xis odd, and aand bare relatively prime, it
follows that x=ab, both aand bare odd, y= (a2−b2)/2, and z= (a2+b2)/2.
The table shows all primitive Pythagorean triples with 1 ≤b < a ≤81. Dots appear in place of the
nonprimitive Pythagorean triples.
In the first box a= 3 and b= 1. In the second box a= 5 and b= 1,3. In the third a= 7 and b= 1,3,5.
Etc.
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