In the following time, Tao Li spent most of his energy on guiding the three people's papers, but all this was based on the premise that there was no problem with the final review of class 18, otherwise Tao Li would not have spent his time in this area upside down.

Although all three of them are new to papers, they are very serious about studying, and their own talents are also very good, so they can quickly grasp some basic knowledge that should be paid attention to in papers. I know all about abstracts, keywords, citations, etc.

"There are actually many ways to calculate the general terms of the Fibonacci sequence, and the more complex ones can be programmed by computer or calculated manually, and I can simply introduce a few methods to you. ”

In an empty classroom on the fourth floor of the third year of high school, Tao Li is teaching Zhang Xingchen, Shen Lei and Zheng Tongya about the Fibonache sequence.

"The most commonly used is the eigenequation method, according to the recursive formula, we let x^2=x+1, then solve the root of the equation, and then bring two eigenvalues into this equation...... Finally, you can get its general term. ”

After Tao Li finished talking about the first method, he put down the chalk in his hand and looked at the three people. This method is actually very simple, not to mention students who have participated in math competitions, but also ordinary students, and the teacher will also talk about this method during class.

"This is the first method, and of course, you can also use the parent function method, which involves combinatorics, but it's nothing new to the three of you. ”

After a pause, Tao Li quickly erased what he had just written, and continued to speak,

"We write out the parent function of this sequence, and by combining the same terms and simplifying it, we can get an exact expression of G(x), and then expand it by power series...... The final result is the same, of course, if you are not afraid of trouble, you can also use linear algebra, but the idea of this method is the same as the characteristic equation, I will not explain much, if you are interested, you can use the method of linear algebra to solve it yourself. ”

After talking about the second method, Tao Li will not say more, about the general formula of the Fibonacci sequence, you can master it here, there is no need to go further.

"I'll let you go back and take a look at the information I gave you in the past few days, you should have read about it, I don't know what you think about the specific direction of this paper?"

Tao Li found a chair and sat in front of the three people, and asked with a smile. So many days of research are enough for them to grasp the basic properties of the sequence, and with an understanding of these properties, it is relatively simple to find out the specific direction of the paper.

"I took it seriously, but I still feel a little confused and don't know where to start. Zheng Tongya was the first to say goodbye.

"I'm in a similar situation to Ago, I don't have a clue. Shen Lei also replied.

"What about you, Xingchen?" Tao Li finally looked at Zhang Xingchen, who was sitting across from him, and asked.

"I had an idea, but I just didn't know if it would work. Zhang Xingchen scratched his head a little embarrassed and said.

"Then you can say it, don't say how we know it's okay. Tao Li said helplessly. Zhang Xingchen is sometimes too modest, not aggressive and leading, otherwise it would be good for him to be the leader of the three.

For example, one of every 3 consecutive numbers is divisible by 2, and only one of every 4 consecutive numbers is divisible by 3, and the information given by the teacher only introduces that there are 9 consecutive numbers and only one is divisible by 34, I am wondering if there is such a general law that can be quantified by a formula, so that only one of the consecutive N numbers is divisible by F(N). ”

Speaking of this, Zhang Xingchen smiled embarrassedly, not knowing if what he said was right.

"Keep talking!" Tao Li's eyes lit up, Zhang Xingchen said that the direction coincided with him, and it was a good thesis topic.

"I thought about this direction for a while, it should be feasible, and the level of the three of us, there should be no problem in asking for such a formula. But I still have a less mature idea, and I don't know if it's going to work. By studying this sequence, we can see that the 5th, 7th, 11th, 13th, 17th, and 23rd digits in the sequence are prime numbers: 5, 13, 89, 233, 1597, 28657, so my idea is, are there infinite numbers in the Fibonacci sequence?"

Speaking of this, Zhang Xingchen directly set his eyes on Tao Li, obviously he couldn't answer this question at the moment, and he couldn't explain it for a while.

"I think the second direction you mentioned is much better than the first, but it will definitely not be possible to come up with a result anytime soon, and at the moment, I can't answer this question. As for the second question about prime numbers, we will study it, and we don't necessarily have to answer the final conclusion, but we can write as much as we can!"

Tao Li looked at Zhang Xingchen with great appreciation, his eyes were full of joy, it seems that the system's evaluation of the three of them is not wrong, Zhang Xingchen is really the most talented one among the three.

"I don't have any problems, I'll do whatever Brother Chen does anyway!" Shen Lei was the first to say.

"Then I naturally followed the big army. Zheng Tongya spread his hands and said that he could do it.

"Well, since the direction of the paper has been determined, then from tomorrow, we will officially enter the writing of the paper, and we will gather in the library in the future, we need to find some relevant papers and books as references, how to search for papers, I will teach you. Tao Li said quickly.

The rest of the time was very fulfilling, every afternoon after school, the four of them would rush into the library, look up materials, take notes, and then divide the work to calculate and verify. At first, Tao Li would go with them, but then it was just the three of them, because Tao Li felt that he didn't have to follow the whole time.

After two weeks, the three of them finally found Tao Li with the first draft of the paper, although they had reached a preliminary conclusion, but after all, it was the first time to write a paper, and there were many errors in the format and writing that they would not notice.

In this way, I revised it three times repeatedly, and the paper written by the three people was finally qualified, not only the repetition rate was only 2.3%, but the academic nature of the paper was also very strong. It's time to deliver the paper.