Primitive Pythagorean Triples on a Curvilinear Grid Defined by Euclid's Formula

scale

0.3

range

124

A primitive Pythagorean triple is a set of values

(a,b,c)

satisfying the Pythagorean theorem

2

a

+

2

b

=

2

c

, with no common factors,

gcd(a,b,c)=1

. The triples can be rewritten using Euclid's formula as

(mn,

2

m

-

2

n

,

2

m

+

2

n

)

with

m

and

n

coprime and

m-n

odd to ensure that the triples are primitive. The primitive triples are represented by their corresponding triangles, scaled and translated to the intersection of the curves for each value

a

,

b

, in the centroid of the triangle. Empty intersections occur where the triple is not primitive.