Parent/child Relationships
By a result of Berggren (1934), all primitive Pythagorean triples can be generated from the (3, 4, 5) triangle by using the three linear transformations T1, T2, T3 below, where a, b, c are sides of a triple:
| new side a | new side b | new side c | |
| T1: | a − 2b + 2c | 2a − b + 2c | 2a − 2b + 3c |
| T2: | a + 2b + 2c | 2a + b + 2c | 2a + 2b + 3c |
| T3: | −a + 2b + 2c | −2a + b + 2c | −2a + 2b + 3c |
If one begins with 3, 4, 5 then all other primitive triples will eventually be produced. In other words, every primitive triple will be a “parent” to 3 additional primitive triples. Starting from the initial node with a = 3, b = 4, and c = 5, the next generation of triples is
| new side a | new side b | new side c |
| 3 − (2×4) + (2×5) = 5 | (2×3) − 4 + (2×5) = 12 | (2×3) − (2×4) + (3×5) = 13 |
| 3 + (2×4) + (2×5) = 21 | (2×3) + 4 + (2×5) = 20 | (2×3) + (2×4) + (3×5) = 29 |
| −3 + (2×4) + (2×5) = 15 | −(2×3) + 4 + (2×5) = 8 | −(2×3) + (2×4) + (3×5) = 17 |
The linear transformations T1, T2, and T3 have a geometric interpretation in the language of quadratic forms. They are closely related to (but are not equal to) reflections generating the orthogonal group of x2 + y2 − z2 over the integers. A different set of three linear transformations is discussed in Pythagorean triples by use of matrices and_linear transformations. For further discussion of parent-child relationships in triples, see: Pythagorean triple (Wolfram) and (Alperin 2005).
Read more about this topic: Pythagorean Triple
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