Primitive Pythagorean Triples 4: Ordered Tree Matrices

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Given a primitive Pythagorean triple (PPT), three new PPTs, which we call children of the original, are generated by multiplying the original on the right by three fixed matrices called Hall matrices. Moreover, any PPT can be generated by a series of such right-multiplications starting from the PPT . Equivalently, we can generate the three new PPTs by right-multiplying "redundant forms" of the original PPT by a single matrix. The PPT has a total of eight redundant forms, , but only the four redundant forms, , are required.

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In this Demonstration, the single matrix is denoted . The " derivation" button shows how the Hall matrices are computed as a combination of and other matrices described in Details, and the " button shows the generation of children using Hall matrices.

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Contributed by: Robert L. Brown (November 2010)
Open content licensed under CC BY-NC-SA


Snapshots


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.



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