Twin and Nearly Twin Pythagorean Triples

A Pythagorean triple is an ordered set of three positive integers that are the side lengths of a Pythagorean triangle, so that . A twin Pythagorean triple is a Pythagorean triple in which two elements differ by one. In a twin leg-leg Pythagorean triple, and differ by 1; in a twin leg-hypotenuse or ; in a nearly twin leg-hypotenuse triple, or .
The twin triples and are called the base twin triples. This Demonstration shows that all twin Pythagorean triples can be calculated by a base twin repeatedly multiplied on the right times by a carefully chosen matrix.

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The (base 3) ID numbering system suggested by the author captures a concise description of the underlying formula for primitive Pythagorean triples. More than that, ID numbers with repeating last digits yield insight to other nearly twin and similar Pythagorean triples.
This method for generating twins also generates two classes of nearly twin leg-hypotenuse Pythagorean triples. Here are some additional examples.
There are no nearly twin leg-leg Pythagorean triples, that is, where ; the next possible integer after for is . Here are a few examples.
For a tool to look up and explore ID patterns, see the Demonstration, "Primitive Pythagorean Triples 3: Ordered Tree Graph".
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