New Method for Solving the Collatz Conjecture 10 with High-Prize Money

Chapter 8: Encounter

　

    Please read "Introduction to a New Method in Chapter 1 (article 1)"

before reading Chapter 8 (article 10).

   Chapter 1 also posts links to each chapter.

   When translating from Japanese to English, subtle nuances of the original articles may not be fully conveyed. If you would like to view the original Japanese articles, please see Chapter 1 of Japanese Article 1.

   Summary Video of the New Method in English

8.1 Encounter

   It's no exaggeration to say that encounters with people, things, and information shape one's destiny and direction in life.

In their 1975 paper on the Li-Yorke theorem, Li and Yorke named the phenomenon
"Period 3 implies Chaos".
This was the first time the term "chaos" was used in a mathematical context.

   The first half of the Li-Yorke theorem states that if a continuous function $${f}$$ on the interval $${[a,b]}$$ satisfies $${a \le f(x) \le b}$$ and has period 3, then $${f}$$ has all periods.
   This is a remarkable theorem, as it implies that if a function has period 3, it must also exhibit all other possible periods.

   The Li-Yorke theorem and the very concept of chaos might not have existed without a chance encounter. Here is an anecdote about how it happened.

   Professor Yorke of the University of Maryland and then-graduate student Li, who was from Taiwan, became interested in why a hump-shaped function (such as the tent map) could generate random sequences. This function was introduced to Professor Yorke by Professor Alan Faller, a geophysics professor. Li and Yorke began to collaborate on what properties this function should have.

   In 1974, Professor Yorke first believed he had found a proof, but he realized it was incorrect. After a few weeks, Li thought he had found a proof, but Professor Yorke found an error in his approach. Li then found a way to correct the error, and they finally arrived at a correct proof. Although they had completed the proof of the Li-Yorke theorem, they did not realize its significance at that time.

   The paper was short, only two or three pages long, and they had not considered the content carefully. As a result, they submitted it to a journal for mathematics educators.

   The editor said that it should be written more gently and sent again, as it is for mathematics educators.

   They were discouraged and lost the motivation to revise it, so they put the paper away on a shelf.

   Half a year had passed. In 1974, Robert May came to the University of Maryland to give a lecture on mathematical biology. Li and Yorke were excited by the results of the numerical experiments presented in the lecture, as they exactly matched the research they had been analyzing.

   As soon as the lecture ended, they met with May and discussed their analysis. May was also surprised by the brilliant analysis.

   That night, they took out the draft of the paper they had put away, added some supplementary information about the proof, and resubmitted the 7-page draft. The paper was accepted and published to the world in 1975.

   The response was overwhelming. The 200 printed copies of the paper were quickly depleted by requests from all over the world.

   The field of chaos was born out of a series of serendipitous events. Alan Turing introduced the function to Lee and Yorke. They attended May's lecture. They were so impressed by the value of their research that they decided to publish it. (These are anecdotes that Yamaguchi heard directly from Lee and Yorke,, as described in his book, Chaos and Fractals)

   Though I cannot compare to them, learning about unsolved historical problems has changed my values considerably. I am no longer easily swayed by those around me.

   Let's use height as an example. When we're standing on the ground, we can see the difference even if it's just a few millimeters. But what if we look from the second floor? We can't see the difference of a few millimeters, but we can see the difference of three or four centimeters. And as we go up to the third floor, the fifth floor, and the tenth floor, the difference becomes less and less visible.

   Historical figures are on higher floors, you see. Einstein is on the 100th floor or something, and Euler, Gauss, and Newton are on the 90th floor or above. We can't meet great people in reality, but we can meet them in our dreams. When we go to the higher floors and look down, people look like ants, so small that we can't see the difference in height and they all look the same. It's the same with great people, isn't it? Lee and Yorke are probably on the 40th floor or so. Even if we look down from there, we can't see the difference in height.

   Coming back from a dream, various things are mixed up in the real world on the ground, such as temporary topics on TV and the internet, hierarchical relationships in the company, and neighborhood relationships.　However, now I have come to see that everyone is working hard, I myself am working hard, and most people, both myself and others, are the same.

   It seems that some people who worked on deciphering prime numbers felt that they "wasted their lives" (maybe scholars?). I don't know what it means to be a waste, but I don't think so. Rather, I think that's what it is.

   In terms of Olympic selection, only one or two athletes (or teams) from each sport can qualify. And out of all the athletes from around the world, only one (or one team) can win the gold medal. That means 99.99% of people will never win a gold medal. But on the other hand, if you don't try, you can't win at all.

The rarer something is, the more valuable it becomes.

　

8.2 Other Unsolved Prize Problems That Seem Challenging

   $${P \neq NP}$$ conjecture is one of the seven Millennium Prize Problems offered by the Clay Mathematics Institute, with a $1 million USD prize for its solution.

   P represents the set of problems that can be solved in polynomial time, meaning they can be solved efficiently by a computer. NP represents the set of problems for which a correct solution can be verified in polynomial time.

   If P equals NP, then all problems in NP can be solved in polynomial time.
It's a conjecture that such a convenient thing is unlikely to happen.

   Even if you don't go head-to-head with the $${P \neq NP}$$ conjecture, you can still win $1 million by finding a "deterministic" algorithm that can solve any one of the following problems in polynomial time:

　　　　1. Traveling Salesman Problem

　　　　2. Clique Problem

　　　　3. Graph Coloring Problem

　　　　4. Hamiltonian Path Problem

　　　　5. Partition Problem

　　　　and so on.

   The $${P \neq NP}$$ conjecture may seem confusing at first, but these problems might make it clearer.

   The traveling salesman problem (TSP) is a classic optimization problem. In this problem, a salesman must visit a set of cities and return to the starting city. The goal is to find the shortest possible route that visits each city exactly once.

   The number of possible routes in the brute-force method is given by $${(n-1)!/2}$$, where n is the number of points. For 30 cities, this number is $${4.42\cdot 10^{30}}$$. Even with the latest computers, it would take 250,000 billion years to calculate all possible routes, which is much longer than the age of the universe (13.7 billion years). Therefore, the brute-force method is not feasible for large problems.

   A solution can be found if a recurrence formula of the form　$${N_{k+1}=h(N_k)}$$ can be identified in some way, instead of using the brute-force method.

   Please also look up the remaining 4 questions.

   Leaving a mark on the history of mathematics is far more valuable than any prize money.