Open AccessConnections on Valuated Binary Tree and Their Applications in Factoring Odd Integers
Author(s) -
Xingbo Wang,
Jinfeng Luo,
Ying Tian,
Li Ma
Publication year - 2021
Publication title -
asian research journal of mathematics
Language(s) - English
Resource type - Journals
ISSN - 2456-477X
DOI - 10.9734/arjom/2021/v17i330287
Subject(s) - connection (principal bundle) , mathematics , binary tree , factorization , integer (computer science) , node (physics) , binary number , discrete mathematics , combinatorics , computer science , algorithm , arithmetic , geometry , structural engineering , engineering , programming language
This paper makes an investigation on geometric relationships among nodes of the valuated binary trees, including parallelism, connection and penetration. By defining central lines and distance from a node to a line, some intrinsic connections are discovered to connect nodes between different subtrees. It is proved that a node out of a subtree can penetrate into the subtree along a parallel connection. If the connection starts downward from a node that is a multiple of the subtree’s root, then all the nodes on the connection are multiples of the root. Accordingly composite odd integers on such connections can be easily factorized. The paper proves the new results with detail mathematical reasoning and demonstrates several numerical experiments made with Maple software to factorize rapidly a kind of big odd integers that are of the length from 59 to 99 decimal digits. It is once again shown that the valuated binary tree might be a key to unlock the lock of the integer factorization problem.