Editing Pythagorean triple (section)

==Parent/child relationships==
{{Main|Tree of Pythagorean triples}}

By a result of {{harvtxt|Berggren|1934}}, all primitive Pythagorean triples can be generated from the (3, 4, 5) triangle by using the three [[linear transformation]]s T<sub>1</sub>, T<sub>2</sub>, T<sub>3</sub> below, where {{math|''a''}}, {{math|''b''}}, {{math|''c''}} are sides of a triple:
{| cellpadding=5 style="text-align:right"
|-
| ||new side {{math|''a''}}||new side {{math|''b''}}||new side {{math|''c''}}
|-
|{{math|T<sub>1</sub>}}:|| {{math|''a'' − 2''b'' + 2''c''}}|| {{math|2''a'' − ''b'' + 2''c''}}||{{math|2''a'' − 2''b'' + 3''c''}}
|-
|{{math|T<sub>2</sub>}}:|| {{math|''a'' + 2''b'' + 2''c''}}|| {{math|2''a'' + ''b'' + 2''c''}}||{{math|2''a'' + 2''b'' + 3''c''}}
|-
|{{math|T<sub>3</sub>}}:|| {{math|−''a'' + 2''b'' + 2''c''}}|| {{math|−2''a'' + ''b'' + 2''c''}}|| {{math|−2''a'' + 2''b'' + 3''c''}}
|}

In other words, every primitive triple will be a "parent" to three additional primitive triples.
Starting from the initial node with {{math|1=''a'' = 3}}, {{math|1=''b'' = 4}}, and {{math|1=''c'' = 5}}, the operation {{math|T<sub>1</sub>}} produces the new triple 
:(3 − (2×4) + (2×5), (2×3) − 4 + (2×5), (2×3) − (2×4) + (3×5)) = (5, 12, 13), 
and similarly {{math|T<sub>2</sub>}} and {{math|T<sub>3</sub>}} produce the triples (21, 20, 29) and (15, 8, 17).

The linear transformations T<sub>1</sub>, T<sub>2</sub>, and T<sub>3</sub> have a geometric interpretation in the language of [[quadratic form]]s. They are closely related to (but are not equal to) reflections generating the [[orthogonal group]] of {{math|''x''{{sup|2}} + ''y''{{sup|2}} − ''z''{{sup|2}}}} over the integers.<ref>{{harv|Alperin|2005}}</ref>