Carry Value Transformation (CVT) - Exclusive OR (XOR) Tree and Its Significant Properties

TL;DR

This paper introduces the CVT-XOR Tree, a recursive structure combining CVT and XOR operations to analyze the sum of non-negative integers, revealing properties like node classification, depth, and connections to Goldbach's Conjecture.

Contribution

It presents a novel tree structure for CVT and XOR operations, analyzing its properties and providing matrices for node information, with implications for classical number theory problems.

Findings

Analysis of parent, leaf, and internal nodes in CVT-XOR Trees

Development of matrices to determine node properties

Insights related to Goldbach's Conjecture

Abstract

CVT and XOR are two binary operations together used to calculate the sum of two non-negative integers on using a recursive mechanism. In this present study the convergence behaviors of this recursive mechanism has been captured through a tree like structure named as CVT-XOR Tree. We have analyzed how to identify the parent nodes, leaf nodes and internal nodes in the CVT-XOR Tree. We also provide the parent information, depth information and the number of children of a node in different CVT-XOR Trees on defining three different matrices. Lastly, one observation is made towards very old Mathematical problem of Goldbach Conjecture.