Trees (data structures) | Diophantine equations
Tree of primitive Pythagorean triples
In mathematics, a tree of primitive Pythagorean triples is a data tree in which each node branches to three subsequent nodes with the infinite set of all nodes giving all (and only) primitive Pythagorean triples without duplication. A Pythagorean triple is a set of three positive integers a, b, and c having the property that they can be respectively the two legs and the hypotenuse of a right triangle, thus satisfying the equation ; the triple is said to be primitive if and only if the greatest common divisor of a, b, and c is one. Primitive Pythagorean triple a, b, and c are also pairwise coprime. The set of all primitive Pythagorean triples has the structure of a rooted tree, specifically a ternary tree, in a natural way. This was first discovered by B. Berggren in 1934. F. J. M. Barning showed that when any of the three matrices is multiplied on the right by a column vector whose components form a Pythagorean triple, then the result is another column vector whose components are a different Pythagorean triple. If the initial triple is primitive, then so is the one that results. Thus each primitive Pythagorean triple has three "children". All primitive Pythagorean triples are descended in this way from the triple (3, 4, 5), and no primitive triple appears more than once. The result may be graphically represented as an infinite ternary tree with (3, 4, 5) at the root node (see classic tree at right). This tree also appeared in papers of A. Hall in 1970 and A. R. Kanga in 1990. In 2008 V. E. Firstov showed generally that only three such trichotomy trees exist and give explicitly a tree similar to Berggren's but starting with initial node (4, 3, 5). (Wikipedia).
