Details

Let be a PPT, that is, , , and are positive integers such that with . A primitive right triangle has sides that are a PPT.

Redundant form means that the PPT definition is relaxed to allow negative values.

Each PPT can be represented by a unique base-3 number called the PPT's ID number. The three children of a PPT have ID numbers formed by appending 0, 1, or 2 to the ID number of the PPT. The parent of a PPT has an ID number formed by dropping the last digit of the PPT's ID number.

Let be a PPT. When post-multiplied by any one of four 3x3 matrices , , , or , a new PPT is the result. These four matrices are:

, , , .

If a suitably selected redundant form of is used, then is the only matrix needed, as shown by clicking the matrix control "P". Also, gives a redundant form of the parent PPT, because .

However, the proper redundant form can be calculated by first multiplying by , , where

, , ;

that is, , , and . Set , , and . These are shown with the matrix control set to " and derivation".

The Demonstration "Primitive Pythagorean Triples 2: Ordered Pairs" shows that adding to each element of accomplishes the same thing as multiplication by .

Define ; then and , where is the 3×3 identity matrix.

Any PPT can be obtained by post-multiplying or by a unique chain of matrices. Their subscripts spell out a unique PPT ID number, as shown in "Primitive Pythagorean Triples 3: Ordered Tree Graph".

MathWorld's "Pythagorean Triple" (Wolfram MathWorld)and other sources use , , and , as in [1].

References

[1] B. Berggren, "Pytagoreiska trianglar" (in Swedish), Tidskrift för elementär matematik, fysik och kemi, 17, 1934 pp. 129–139.

[2] A. Hall, "Genealogy of Pythagorean Triads," Mathematical Gazette, LIV(390), 1970 pp. 377–379.

[3] Formulas for generating Pythagorean triples

[4] H. Lee Price, "The Pythagorean Tree: A New Species", 2008 http://arxiv.org/abs/0809.4324.