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If one begins with primitive triple [3,4,5], then all primitive triples will eventually be produced by application of these matrices. The result can be graphically represented as an infinite ternary tree with [ a , b , c ] at the root node.
A primitive Pythagorean triple is one in which a, b and c are coprime (that is, they have no common divisor larger than 1). [1] For example, (3, 4, 5) is a primitive Pythagorean triple whereas (6, 8, 10) is not. Every Pythagorean triple can be scaled to a unique primitive Pythagorean triple by dividing (a, b, c) by their greatest common divisor ...
A tree of primitive Pythagorean triples is a mathematical tree in which each node represents a primitive Pythagorean triple and each primitive Pythagorean triple is represented by exactly one node. In two of these trees, Berggren's tree and Price's tree, the root of the tree is the triple (3,4,5), and each node has exactly three children ...
The set of all nodes at a given depth is sometimes called a level of the tree. The root node is at depth zero. Height - Length of the path from the root to the deepest node in the tree. A (rooted) tree with only one node (the root) has a height of zero. In the example diagram, the tree has height of 2. Sibling - Nodes that share the same parent ...
A Pythagorean triangle is right-angled and Heronian. Its three integer sides are known as a Pythagorean triple or Pythagorean triplet or Pythagorean triad. [9] All Pythagorean triples (,,) with hypotenuse which are primitive (the sides having no common factor) can be generated by
[4] [6] The first three of these define the primitive Pythagorean triples (the ones in which the two sides and hypotenuse have no common factor), derive the standard formula for generating all primitive Pythagorean triples, compute the inradius of Pythagorean triangles, and construct all triangles with sides of length at most 100. [6]
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In elementary arithmetic geometry, stereographic projection from the unit circle provides a means to describe all primitive Pythagorean triples. Specifically, stereographic projection from the north pole (0,1) onto the x -axis gives a one-to-one correspondence between the rational number points ( x , y ) on the unit circle (with y ≠ 1 ) and ...