QUADRUPLETS

Abstract

In this paper we treat the one of the most known and popular topics. In 1963, Barning discovered a ternary tree with nodes the Primitive Pythagorean triplets by using 3 matrices that helped him propagate through the tree. Independently, Hall found the same thing after seven years. In this paper we go through the ways that we know to generate triplets and try them to help us find another tree, this time a little bit different based on a new conjecture. A new way to find triplets also is found which we programmed to find quadruplets but with slight modification of the code can also find the matrices found by Barning and Price. We also try another method using matrix transformations. We use matrices even though they are inefficient from computation time point of view but they are an elegant representation and useful to observe patters. During our work with the topic, interesting results are found related to the tree of Barning and Price which we have stated them at the results. Promising results are also found related to the quadruplets tree. Some matrices that help us propagate 3 steps are found and give an insight of how the tree may be formed.