The Path to Resolve the Cmi Millennium Problems

By Shi Feng Sheng and Danny Wong

This book is not for everyone, but a must for researchers in the field of number theory, topology, computer science and physics, or anyone (loves mathematics and science) with college level knowledge, curious spirit and an open mind.

Proclaimed solution of the 1742 Goldbach’s conjecture by Mr. Shi proved the principal problem in number theory was “arithmetic” in nature, together with the other topics addressed in his book --- illustrated the mathematical knowledge is not a collection of isolated fact. Each branch is a connected whole; linked to other branches that we do not understand mathematically, but ultimately, they are all connected to the roots of mathematics: the pattern of the primes.

Moreover, we are optimistic solution of the CMI problems and other conundrums addressed in this book were credible because --- nothing occurs contrary to nature except the impossible, and that never occurs (Galileo 1564 -1642).

• Language: English
• Publisher: AuthorHouse
• Release date: Apr 14, 2022
• ISBN: 9781665553360

Shi Feng Sheng

Author Shi Feng Sheng is a retired naval architect/engineer; but in 1994 --- Education Department of the National Geographic Society believed he was the number one mathematician in the world after he read his unpublished advanced mathematical book copyrighted with the Library of Congress. Title: EXIST/ Sheng, Shi Feng, with registration & date TXU00046507311990-12-04. This book addressed (six) famous unsolved mathematical problems ranging from ancient Greeks (BC) to old problems in number theory (1637, 1742) Co-author Danny Wong is a retired businessman, an amateur number theory enthusiast and assistant to Mr. Shi Feng Sheng. Wong’s years of exhaustive researches led them to conclude that--- the topics addressed in EXIST were the essential ingredients needed to resolve all the CMI millennium prize problems.

Book preview

THE PATH TO

RESOLVE

THE CMI MILLENIUM PROBLEMS

Shi Feng Sheng and Danny Wong

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ACKNOWLEDGEMENT

THE PATH TO

RESOLVE

THE

CMI MILLENNIUM PROBLEMS

This Book

Would Not Be Possible

Without

The Valuable Insight Of A Book

Entitled

The Millennium Problems

By Keith Devlin

© 2022 Shi Feng Sheng and Danny Wong. All rights reserved.

No part of this book may be reproduced, stored in a retrieval system, or transmitted by any means without the written permission of the author.

AuthorHouse™

1663 Liberty Drive

Bloomington, IN 47403

www.authorhouse.com

Phone: [phone]

Because of the dynamic nature of the Internet, any web addresses or links contained in this book may have changed since publication and may no longer be valid. The views expressed in this work are solely those of the author and do not necessarily reflect the views of the publisher, and the publisher hereby disclaims any responsibility for them.

ISBN: 978-1-6655-5335-3 (sc)

ISBN: 978-1-6655-5336-0 (e)

Library of Congress Control Number: 2022903806

Published by AuthorHouse 06/15/2022

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Contents

PREFACE

ONE Early History In Mathematics (BC to 19th century)

TWO Recent History In Mathematics (20th to 21st century)

The 1900 David Hilbert’s twenty-three problem

The 1989 unpublished Advanced Mathematical Book entitled Exist

Copyrighted with the Library of Congress

Registration Number & Date: TXU0003465073/1990 – 12 – 4

Goldbach’s Conjecture

An infinite Calculation Chart (reading instruction)

Fermat’s Last Theorem

Trisecting an Angle

Heptagon

Doubling the Cube

Squaring the Circle

The 2000 CMI Millennium Prize Problems

P vs. NP (P = NP) and (P ≠ NP)

Riemann Hypothesis

Birch and Swinnerton – Dyre Conjecture

Poincare Conjecture

Hodge Conjecture

The Yang-Mills Quantum Theory

The Naiver-Stokes Equations

THREE Other Intractable Problem In Mathematics

Twin Primes Conjecture

Beal’s Conjecture

Construct a 17-sided polygon

FOUR Appendix

A: Six letters of acknowledgment

A-1: Pattern of the odd primes (in a reasonable range)

B: Diagram of trisecting the Angle

B-1: Diagram of Heptagon

C: Diagram of Doubling the Cube

C-1: Diagram of Squaring the Circle

D: A graphic proof of the (GC) and (Twin primes)

D-1: A color diagram

E-1: An infinite color calculation chart (17x 11)

E-2: Structure of chart E-1(17 x 11)

PREFACE

[1] December 2009 was truly an extraordinary month in my life:

While dropping a friend off in downtown St Petersburg Florida; I met a distinguished looking elder Chinese (Mr. Shi). As we talked, somehow our conversation led him to claim that he had resolved the famous 1742 Goldbach’s Conjecture (the principal unsolved problem in pure mathematics) originated from St Petersburg Russia as well as many other famous intractable problems in mathematics --- which I heard of during college years.

Both of us were amazed because: (a) it is very rare to see Asians in downtown St Petersburg, let alone to meet someone who speaking the same Shanghai dialogue, (b) Mr. Shi was presently surprised that I knew of his mathematical problems, and I was surprised of what he proclaimed, but skeptical. Nevertheless, we agreed to meet again soon.

Coincidently, my son (Timothy Wong), an amateur number theory enthusiast gave me a Math book entitled The Millennium Problems authored by Keith Devlin (Stanford) for Christmas a week earlier --- this book consists of seven intractable problems in the field of number theory, topology, physics and computer science selected by the Clay Mathematical Institute. Moreover, CMI asks for their solutions with prize of $1.000.000 per/problem --- subject to certain constrains.

[2] When we met again the day after Christmas, Mr. Shi presented me with:

* A manuscript of his math book entitled EXIST --- which addressed two major unsolved problems in number theory namely; the 1742 Goldbach’s conjecture, the 1637 Fermat’s Last Theorem, and four classic Greek problems from (2000+ BC) --- trisecting the angle, heptagon, doubling the cube and squaring the circle.

* Diagram of a (17-sided polygon) and claimed it was done by the first human (his high student) from Paris France --- via Straightedge and Compass construction.

Undoubtedly, there will be serious ramification in the mathematics and beyond if the addressed topics in his Book and the 17-sided polygon turned out to be credible.

[3] Months of tireless clarification of his texts (some were vague) in EXIST and the valuable in-sight of the subjects in "The Millennium Problems" --- renewed my curious spirit in mathematics --- to commit years of investigation, and concluded that the topics in EXIST were directly or indirectly linked to all the CMI problems; their connections are best understood to present them together in the (form) of this book because:

(a) With respect to his (six) topics addressed in EXIST: (i) papers on the Goldbach’s conjecture and the Fermat’s Last Theorem by famous mathematicians were already published by reputable Journals as a-step-in-the-right-direction because they were consistent with existing research activities, (ii) other four problems of Antiquity had declared by reputable mathematicians as insoluble a long time ago, (iii) with respect to the CMI problems, they were based on modern abstract theories, theorems and conjectures --- that was (is) unintelligible even to the experts in the field.

Consequently, there are fair bits of upstream background to cover before I can begin to re-introduce the topics in EXIST, and lots of downstream abstractions to uncover before I can elaborate the CMI problems from our perspective.

(b) The (proclaimed) need to be studied together in the sequence we presented in order to illustrate that --- mathematical knowledge is not a collection of isolated facts. Each branch is a connected whole; linked to other branches that we do not understand mathematically, but ultimately, they are all connected to the roots of mathematics: the pattern of the primes.

(c) Due to the time consuming vetting rules stipulated by the CMI --- we decided that there is no time to waste because professor Shi is (87 years old); not mention that our proclaimed were totally inconsistent with the formal, the symbolic, the verbal, the analytic elements and modern abstract theories --- passed down by the famous or not-so-famous researchers. Nevertheless, this book is consistent with --- the nature mathematics (or arithmetic) passed down from the ancient Greeks and Fibonacci (1170-1240) and the famous quotes left by the great Galileo (1564-1642):

It is surely harmful to souls to make a heresy to believe what is proved.

Mathematics is the language with which God has written the universe.

We must say that there are as many squares as there are numbers.

Where the sense fails us, reason must step in.

Truths are easy to understand once they are discovered; the point is to discover them.

Nothing occurs contrary to nature except the impossible, and that never occurs.

More importantly, to illustrate the classic mechanics developed by Newton (1642-1726) was a precursor of the tremendous advancement in modern science and technology of yesterday and today.

Danny Wong

Sarasota Florida

January 2022

E-mail: [e-mail]

ONE

Early History In Mathematics (BC to 19th century)

God created the universe and natural numbers, everything else is made by men. Infinite universe and natural numbers; macroscopic in distance, microscopic up close.

Prime numbers are any integers ≥ 2 that can be divided by 1 and itself only (two divisors), so 2 is the only even prime. Historically, there is no useful formula that yields all primes and no composites --- because they are not polynomial.

Historically, mathematicians have: (1) dealt with questions of finding and describing the intersection of algebraic curves, (2) wrestled with paradoxes of the pattern of the prime numbers, concept of infinite and sum of the infinite series long before --- Euclid (300 BC) devoted part of his Elements to prime numbers and divisibility, topics the belong unambiguously belong to number theory; and introduced the 1st proof of infinitude of primes by abstract reasoning, the Euclidean geometry and considered line, circle as curves in geometry; the work of Archimedes (250 BC) and Sieve of Eratosthenes (240 BC).

In the 3rd century Arithmetica introduced Diophantine geometry --- a collection of problems giving numerical solutions of both determinate and indeterminate equations. Diophantine studied rational points on curves (elliptic) and algebraic varieties. In other words, Diophantine showed how to obtain infinitely many of the rational numbers satisfying a system of equations by giving a procedure that can be made into as an algebraic expression (algebraic geometry in pure algebraic forms). Diophantus contributed greatly in mathematical notation, and introduced approximate equality to find maxima for functions and tangent line to curves. Unfortunately, most of the texts were lost or unexplained.

Some of the roots of algebraic geometry date back to the work of Hellenistic Greeks (450 BC). The Delain problem --- doubling the cube and other related problems such as; trisecting the angle, polygons, squaring the circle --- they are also known as straightedge and compass problems (or topological problems).

In Geek mathematics --- Number theory (or arithmetic) is to study the patterns of the numbers and elementary calculation technique, geometry is a technique to study patterns of shape, algebra is to study the patterns of putting things together, and trigonometry considers the measurement of shapes. Topology is to study the patterns of closeness and relative position. Algebraic geometry is to study geometry via algebraic curves and varieties. Not very much happened in mathematics after the Greeks until the 17th century:

* Galileo (1564-1642) was the father of observational astronomy, modern science and a polymath in the field of mathematics, physics, engineering and natural philosophy.

In mathematics --- Galileo applied the standard (arithmetic) passed down from ancient Greeks and Fibonacci (1170-1240) but superseded later by the algebraic methods of Descartes.

* Rene’ Descartes (1596-1630) made a fundamental discovery: Assuming by restrict ourselves to the straightedge and compass in geometry, it is impossible to construct segments of every length. If we begin with a segment of length 1, say, we can only construct a segment of another length if it can be expressed using integers, addition, subtraction, multiplication, division and square roots (as the golden ratio can). Thus, one strategy to prove that a geometric problem is unsolvable (not constructible) --- is to show that the length of some segment in the final figure cannot be written in this way. But doing so rigorously required the nascent field of algebra […]

Descartes introduced the analytic geometry primarily to study algebraic curves to reformulate the (classic works on conic and cube); using Descartes’ approach, the geometric and logical arguments favored by the ancient Greeks for solving geometric problems could be replaced by doing algebra (algebraic geometry extended the mathematical objects to multidimensional and Non-Euclidean spaces).

* Pierre de Fermat (1601-1665) independently developed the analytic geometry to study