Scholar's Advanced Technological System
chapter-228

Chapter 228: Your Hands Are Your Tools

“Neither?”

Molina was stunned.

She looked at Lu Zhou and said with a skeptical tone, “I know you are a genius… Although Goldbach’s conjecture isn’t in my field of research, if I heard you correctly, you’re intending on doing a century worth of work on your own?”

Lu Zhou smiled coldly and said with a relaxed tone, “The problem of a+b is a complex expression of Goldbach’s conjecture. That is, each large even number N can be expressed as A+B, where the prime factors of A and B do not exceed a and b, respectively. When a=b=1, the problem will eventually return to the original expression. Any even number greater than 2 can be written as the sum of two prime numbers.”

One prime factor, naturally meant that it was a prime number.

Therefore, the form of 1+1 was the ultimate form of Goldbach’s conjecture.

Molina said, “So you’re saying that the people who have researched the Goldbach’s conjecture for over a century has been doing nothing?”

“Of course not,” said Lu Zhou as he shook his head. He then asked an unexpected question, “Do you know about sports?”

Molina frowned and said, “Sports?”

Lu Zhou, “You know about the long jump right.”

Molina was confused but she said, “Of course.”

Lu Zhou smiled coldly and said, “Brown’s a+b proof method is equivalent to the run-up before the long jump. Although the run-up time itself is not included in the score, is the run-up useless? The same logic applies here where a+b is equivalent to the run-up of Goldbach’s conjecture. Because without it, there will be no large sieve method, which is an inspiring and potential analytical tool for number theory. It can even be said that the value of the large sieve method is beyond the Goldbach conjecture itself.”

Whether or not the large sieve method could really reach 1+1, it had already played an important role in number theory.

Lu Zhou had personally benefited from it.

Molina brushed her hair as she looked at Lu Zhou and asked, “So, how do you plan on proving it?”

Lu Zhou smirked, “Of course, to use my own method.”

Molina did not know why, but her heart skipped a beat when she saw Lu Zhou’s smile.

Of course, it was only for a second. As a woman married to mathematics, she quickly returned to normal.

…

A solution to a mathematics conjecture required accumulation of workload and a creative genius.

Both were indispensable.

Just like Fermat’s last theorem.

When the Taniyama-Shimura theorem was proved, people could not see the whole picture of the theorem’s value, but they had a rough idea in their minds. This was because a tool to solve the problem had surfaced. This was the historic work by Andrew Wiles.

As for Goldbach’s conjecture, whether it was the large sieve method or circle method, it was the same.

The work of the predecessors built the foundation. However, whether it was Chen’s theorem or the proof of the Goldbach’s conjecture under odd conditions, they were all one step off. The meaning of Chen’s theorem was more to let other mathematicians know that the road of the large sieve method had ended and that there was nowhere else to go.

The circle method was the same.

This was why last year, Helfgott said that “to fully prove Goldbach’s conjecture, we have a long way to go”. He expressed that there was no hope solving Goldbach’s conjecture anytime in the near future.

At least, no hope toward the circle method.

Lu Zhou could not help but agree that both of these methods were at a dead end.

He had also faced similar problems when studying the twin prime conjecture.

Zhang Yitang’s research selected a clever lambda function, which limited the space of prime pairs to 70 million. The successor reduced this number to 246. However, they could not go any further.

Lu Zhou’s initial thought process was also to use a lambda function. However, after countless attempts, he discovered that this road was a dead end.

There were too many lambda function forms to choose from. He could not find the right one no matter how hard he tried.

Until finally, he was inspired. He tried a very different proof of the conjecture and introduced a topology method. This paved a new road.

Although this method was first mentioned in the 1995 thesis by Professor Zellberg who was tackling Goldbach’s conjecture, it was Lu Zhou that introduced it to the problem of prime numbers.

Lu Zhou then built on his own knowledge of group theory and pushed the prime number finite distances to infinity. This solved the Polignac’s conjecture. The topology sieve method had been transformed twice, and completely unrecognizable from its original form.

Therefore, Lu Zhou gave his weapon a new name, “Group Structure Method”.

However, when he was studying the Goldbach’s conjecture, he habitually forgot about his own tools.

On the surface, it seemed that the Group Structure Method was unrelated to Goldbach’s conjecture. However, the intention of the sieve method was to solve Goldbach’s conjecture.

As long as he improved on it, he could use this tool to solve Goldbach’s conjecture.

When a mathematical method was continually perfected, it would transform from a toothpick to a Swiss army knife. It would gradually evolve into a theoretical framework! The theoretical framework for number theory!

This was just like the “Cosmic Teichmüller Theory” created in the study of the ABC conjecture.

Whether it was to develop new methods and then prove the value of the methods or to develop methods while studying the problem, both paths were valid.

Lu Zhou saw hope in Goldbach’s conjecture.

…

Lu Zhou walked out of the food club. However, he did not go to the library. Instead, he went to the Princeton Institute for Advanced Study.

Although he did not make an appointment, Professor Deligne had said that every evening from 6 p.m to 8 p.m. was office hours.

Lu Zhou knocked on the door before he walked in.

Professor Deligne stopped writing and looked at Lu Zhou. He asked in a relaxed tone, “You’ve made a decision?”

Lu Zhou nodded, “Yes, I plan on doing my own research… I apologize but I can’t extract any energy to join your research.”

Deligne nodded and did not show signs of dissatisfaction.

Deligne was a person that respected freedom. That was why he allowed Lu Zhou to make his own decision.

Deligne, “I respect your decision. But as your supervisor, I have to know what your research is about?”

Lu Zhou answered, “Goldbach’s conjecture.”

Deligne nodded. He was not as surprised as Molina. His facial expression was calm.

Maybe…

Deligne thinks that I am the “best candidate” to solve this conjecture?

Thanks for the compliment.

Lu Zhou felt a little proud.

Deligne, “The Goldbach’s conjecture is an interesting problem, I also studied it when I was young. However, I didn’t dive deep into the problem, so I can’t give you much help. Right now the closest research results are Chen’s theorem and Helfgott’s proof of the weak conjecture. I look forward to your new research… ”

“… Of course, other than your own research, there are also some things on my side you have to do. Like teaching assistant work.”

Lu Zhou nodded, “No problem… I’m confident in my teaching abilities on number theory and functional analysis.”

“I believe in your abilities in number theory. In fact, you are overqualified… Also, I prepared a gift for you.”

Deligne pulled out the drawer and took out a certificate looking thing. He then placed it on the table and smiled.

“I heard that your family conditions aren’t good. I helped you solve the problem of your student aid. Take this thing to the finance office, and sort out your tuition fees.”