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

2021 ◽  
pp. 14-34
Author(s):  
Xingbo Wang ◽  
Yuequan Jin
Keyword(s):  
Binary Tree ◽  
Numerical Experiments ◽  
Mathematical Reasoning ◽  
Source Codes

This paper continues investigating connections on a valuated binary tree. By defining three types of new connections, the paper derives several new properties for the new connections, and proves that odd integers matching to the new cases can be easily and rapidly factorized. Proofs are presented for the new properties and conclusions with detail mathematical reasoning and numerical experiments are made with Maple software to demonstrate the fast factorization by factoring big odd composite integers that are of the length from 101 to 105 decimal digits. Source codes of Maple programs are also list for readers to test the experiments.

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

2021 ◽  
pp. 134-153
Author(s):  
Xingbo Wang ◽  
Jinfeng Luo ◽  
Ying Tian ◽  
Li Ma
Keyword(s):  
Binary Tree ◽  
Numerical Experiments ◽  
Mathematical Reasoning ◽  
Binary Trees ◽  
Integer Factorization ◽  
Parallel Connection ◽  
Central Lines ◽  
Factorization Problem ◽  
Integer Factorization Problem

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.

scholarly journals More on Symmetric Brothers of a Node in a Perfect Binary Tree

2021 ◽  
pp. 13-26
Author(s):  
Xingbo Wang
Keyword(s):  
Binary Tree ◽  
Mathematical Reasoning ◽  
Extensive Study ◽  
Blind Search

The paper makes an extensive study on the symmetric brothers of a node in a perfect binary tree. Through proving several new properties of the symmetric brothers of a node, it reveals how the symmetric brothers and the symmetric ancestors distribute on the tree and how they are beneficial for designing a searching algorithm of special purpose. Detail mathematical reasoning and proofs are shown together with concrete examples to demonstrate the mathematical traits. The paper is helpful for designing algorithms in blind search related aspects.

scholarly journals Interval Tree and Its Application in Integer Factorization

2019 ◽  
Vol 11 (2) ◽  
pp. 103
Author(s):  
Xingbo WANG
Keyword(s):  
Binary Tree ◽  
Numerical Experiments ◽  
Simple Approach ◽  
Binary Trees ◽  
Integer Factorization ◽  
Interval Tree

The paper first puts forward a way to study odd integers by placing the odd integers in a given interval on a perfect full binary tree, then makes an investigation on the odd integers by means of combining the original properties of the integers with the properties of the binary trees and obtains several new results on how an odd integer's divisors distribute on a level of a binary tree. The newly discovered law of divisors' distribution that includes common divisors between two symmetric nodes, genetic divisors between an ancestor node and its descendant node can provide a new and simple approach to factorize odd composite integers. Based on the mathematical deductions, numerical experiments are designed and demonstrated in the Maple software. All the results of the experiments are conformance to expectation and validate the validity of the approach.

A Heuristic Algorithm for the Inner-City Multi-Drop

2013 ◽  
pp. 274-293
Author(s):  
Li Pan ◽  
Sydney C. K. Chu ◽  
Guangyue Han ◽  
Joshua Zhexue Huang
Keyword(s):  
Inner City ◽  
Heuristic Algorithm ◽  
Binary Tree ◽  
Numerical Experiments ◽  
Decomposition Approach ◽  
Strong Stimulation ◽  
Tree Data ◽  
Loading Problem ◽  
Tree Data Structure ◽  
Multiple Clients

Economic globalization, increasing fuel cost, and environmental problems provide a strong stimulation for inner-city container carriers to utilize container space more efficiently in transporting goods for multiple clients during a single round trip. A wall-building heuristic algorithm based on the binary tree data structure is proposed to solve the container loading problem with multi-drop constraints. A dynamic space decomposition approach, together with a repacking and space amalgamation strategy, permits an efficient and effective loading plan to pack containers, illustrated by numerical experiments.

Distinct neural substrates recruited for logical and mathematical reasoning

2001 ◽  
Author(s):  
James K. Kroger ◽  
Jonathan D. Cohen ◽  
Philip N. Johnson-Laird
Keyword(s):  
Mathematical Reasoning ◽  
Neural Substrates

Analisis Kemampuan Penalaran Matematis Mahasiswa Calon Guru Matematika Sebagai Kajian Evaluasi

2019 ◽  
Vol 2 (2) ◽  
Author(s):  
Hanifah Nurus Sopiany
Keyword(s):  
High School Students ◽  
Secondary School ◽  
Mathematical Reasoning ◽  
Mathematics Learning ◽  
Good Reason ◽  
School Level ◽  
Real Analysis ◽  
Reasoning Ability ◽  
School Students ◽  
Secondary School Level

Penalaran matematis menggunakan pola pikir logis dalam menganalisa suatu masalah yang nanti pada akhirnya akan ditandai dengan aktivitas menyimpulkan atas masalah tersebut. Seseorang yang memiliki penalaran yang baik, tentunya akan berhati-hati dalam bertindak dan memutuskan sesuatu. Materi-materi pada kalkulus merupakan materi yang ada pada tingkat sekolah menengah yang nantinya menjadi lahan mengajar mahasiswa calon guru matematika S-1. Kemampuan penalaran yang dikaji mempengaruhi pembelajaran mahasiswa kedepannya karena berlaku pada matakuliah lanjut, contohnya pada kemampuan pembuktian akan selalu digunakan pada matakuliah persamaan diferensial, struktur aljabar, analisis  vektor, analisis real, dll. Sedangkan sebagai calon guru yang nantinya mengajar pada tingkat sekolah menengah, maka kemampuan penalaran ini menjadi salah satu capaian pembelajaran matematika bagi siswa sekolah menengah, maka oleh karena itu guru yang mengajarnya haruslah memiliki kemampuan penalaran yang baik. Analisis kesalahan sangat penting untuk melakukan evaluasi dan refleksi pada struktur soal maupun pada perlakuan dalam pembelajaran dalam upaya memperbaiki kemampuan penalarannya.   Mathematical reasoning uses a logical mindset in analyzing a problem that will eventually be marked by concluding activity on the problem. Someone who has good reason, will certainly be careful in acting and deciding something. The material content on the calculus is the material that exists at the secondary school level which will become the field of teaching the prospective master of math teacher bachelor. The reasoning ability studied influences student learning in the future as it applies to advanced courses, for example in the ability of proof will always be used in the course of differential equations, algebraic structure, vector analysis, real analysis, etc. While as a teacher candidate who will teach at the secondary school level, then this reasoning ability becomes one of the achievements of mathematics learning for high school students, therefore teachers who teach it must have good reasoning ability. Error analysis is very important to evaluate and reflect on the problem structure as well as on the treatment in learning in order to improve the reasoning ability.

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