Connections on Valuated Binary Tree and Their Applications in Factoring Odd Integers

Abstract

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.

More Connections on Valuated Binary Tree and Their Applications in Factoring Odd Integers

“…Referring to paper [1], it is seen that three corollaries were proven and those corollaries were quite like the corollaries proven in 4.2. Nevertheless, there are differences.…”

“…This paper follows the study of paper [1], continues the investigation on the connections and their applications in factorization of odd composite integers. By defining three new types of the connections, the paper reasons and obtains several new properties and corollaries in analyzing the odd integers.…”

“…The terms related with the valuated binary tree, subtree, root, node, son, father and ancestors as well as symbols used in this paper can be referred in [1]. Some cited lemmas were also seen in [1].…”

“…The concept of the connection was introduced in [1]. This section mainly studies the connections parallel to the borders of a subtree.…”

“…By means of defining parallelism, connection and penetration, paper [1] investigated geometric relationships among nodes on a valuated binary tree, and it proved several properties about the connections and the…”