Wolfram Demonstrations Project

This Demonstration plots primitive Pythagorean triples (PPTs) to make lovely, interesting patterns in the

plane. The details provide definitions to classify all triples

that are solutions of the Pythagorean formula

in integers. The 2D plot shows integer points

that satisfy this equation for some integer

Let

be a Pythagorean triple (PT), that is,

, and

are positive integers such that

. A primitive Pythagorean triple (PPT) is a PT with

. Redundants are solutions that allow

, and

to be negative. This Demonstration illustrates all 16 solutions,

and

PPTs can be given unique base-3 ID numbers. See the Demonstration "Primitive Pythagorean Triples 3: Ordered Tree Graph" for the definitions.

Click "PPT 1" or "PPT 2" to see the subset of PPTs with base-3 ID numbers beginning with 1 or 2.

If we interpret the plotted

locations as Gaussian integers, then selecting the checkbox

plots the square root of those locations. This square root mode shows why Euclid's formula only appies to "PPT 1" primitive Pythagorean triples.

Contributed by: Robert L. Brown

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Primitive Pythagorean Triples 1: Scatter Plot