From Almighty Scholar to Chief Scientist

[Proof: Suppose V is an n-dimensional linear space, ε1, ε2, ..., εn are a set of basis.Let ψ be the linear transformation of V, so that the matrix of ψ under the basis ε1, ε2, ..., εn is F, that is, ψ(ε1, ε2, ..., εn) = (ε1, ε2, ..., εn) F .

Because F is a friend matrix, so ε2=ψ(ε1), ε3=ψ^2(ε1),..., εn=ψ^(n－1)(ε1). Namely...】

Seeing Lin Xiao's two operations in front of him, Xu Hongbing took a deep breath. Can he use linear space and linear transformation so proficiently?

Even in advanced algebra, this knowledge is also a difficult point. He used to be cautious when asking such questions, for fear of confusing the students.

Moreover, he didn't expect Lin Xiao to think of using this method to solve it so quickly. Generally speaking, students basically use standard unit column vectors to solve it.

He couldn't help feeling in his heart, this student is really amazing.

No wonder being able to show such momentum in the Olympiad, this is simply born for mathematics.

With waves in his heart, he just looked at Lin Xiao to write like this.

It didn't take long for Lin Xiao to write to the last step.

【所以C=（n∑i=1）ci1＊F^（i－1）（e1，Fe1，F^2e1，……，F^（n－1）e1）=（n∑i=1）c1F^i－1

So C is a polynomial in F.

Certificate completed. 】

After writing the last two characters stroke by stroke, Lin Xiao put down the pen.

Seeing that he finished writing, Xu Hongbing said, "Very good, you did a good job."

Lin Xiao smiled and said, "Actually, it's not too difficult, it's nothing."

Xu Hongbing: "..."

"You don't think it's too difficult?"

Lin Xiao nodded, and said truthfully: "It's really okay. I actually just finished reading advanced algebra a few days ago. If I can do it, it shouldn't be too difficult, right?"

Xu Hongbing: "..."

He felt the contempt from the student in front of him for the question he posed.

Moreover, Lin Xiao actually said that he had only finished reading advanced algebra for a few days?

"You just read the high math?"

"Yes."

"Self-study?"

"Self-taught."

"No teacher to guide you?"

"No." Lin Xiao said: "I don't have this kind of college mathematics tutoring in my hometown. Before I participated in the Olympiad, I didn't participate in any training. Oh, that is, before the mathematics league, our school held a month of mathematics league training. class."

Hearing Lin Xiao's words, Xu Hongbing felt even more unbelievable. Lin Xiao is still self-taught?

He originally thought that Lin Xiao was able to do it because of his teacher and his own talent.

Perhaps the teachers of mathematical geniuses have little contribution to their achievements, but mathematical geniuses must have a teacher's guidance.

Newton was so powerful, his teacher was the 'Professor Lucas' of Cambridge University at that time. Professor Lucas did not refer to a certain person, but an honorary position. Only recognized scholars can obtain this position, and at the same time only Being held by a teacher is enough to show how powerful Newton's teacher is.

Besides, Einstein's teacher is Minkowski. The famous Minkowski space-time has a very important position in the field of mathematical physics, and it has a very close relationship with Einstein's later theory of relativity.

Therefore, Xu Hongbing found it incredible.

Lin Xiao lived in such a lack of educational environment, but he still showed such a strong mathematical talent.

Sure enough, gold always shines. At the same time, he couldn't help but feel sorry for Lin Xiao. If he could have a good teacher to guide him, Lin Xiao would have soared into the sky long ago, right?

For example, I joined the IMO team when I was in junior high school.

But soon, he thought that Lin Xiao would be entering university soon, and as expected, Lin Xiao would choose those two schools.

In this way, he doesn't have to worry about Lin Xiao's future. Such a genius will never be ignored in college.

Of course, the above are all the attitudes that he, as a teacher, maintains towards any student and wants him to be good. Leaving aside these things, the questions he asked were underestimated. How can this be tolerated?

Can he be wronged? As a professional mathematics proposition person.

He was invited to participate in the CMO and the proposition group of the college mathematics competition, which is why he was invited to tutor these students.

So he said: "Then you are self-taught, very good, very good, I believe that after you go to university, you will be able to achieve better growth."

"How about this, I will give you a few more difficult questions later, and when you plan to leave later, I will come to the podium to take them."

"Ok."

Lin Xiao nodded.

Chapter 41 Accidentally came up with a problem in the Dao world

Of course, it would take a certain amount of time for Teacher Xu to come up with a question, so Lin Xiao continued to read his book.

Time passed quickly, and at around nine o'clock in the evening, Lin Xiao checked the time, and when it was time to return to the hotel, he stretched himself, then began to pack up the table, ready to go back.

When he walked to the door, he remembered that Teacher Xu said that he would ask himself a few difficult questions, but Teacher Xu seemed to have left the study room just now because of something, and he hasn't come back yet.

However, he should have already worked out the question, right?

So Lin Xiao went to the podium and took a look. There were indeed a few pieces of paper with questions written on them.

In addition, there is a pile of draft paper, which is full of various formulas, quite a lot, but many draft papers are marked with a big cross.

"Ms. Xu, have you encountered any problems?"

Lin Xiao thought about it for a while, but then he didn't worry about it anymore, and picked up the few draft papers with questions written on them.

He briefly glanced at the first sheet of paper. There were questions on both sides. There were four questions in total, and they were basically advanced algebra questions, including determinants, matrices, and other categories.

Then he quickly looked at the next two sheets of paper. The second sheet also had four questions, but the third sheet had only one question. Lin Xiao didn't care what the question was, as long as it wasn't a manuscript paper.

Then he found another piece of paper from the side, and wrote on it [Mr. Xu, I took the questions you asked, and I will come to you tomorrow night—Lin Xiaoliu].

Then, he left the classroom.

Not long after, Xu Hongbing came back. He walked to the podium, saw the notes left by Lin Xiao, and smiled slightly. He racked his brains to come up with those questions, including some questions he had given to math competitions before. .

He wanted to see if Lin Xiao could write it down.

He even looked forward to tomorrow night when this boy would come and cry to him that it was too difficult to do it.

Shaking his head and not thinking about it, he turned up the desk again.

It's not like he has nothing to do when he stays here. As a mathematics professor, he is also researching some problems.

For example, recently, he is studying a world-class problem, that is, whether there are infinitely many prime numbers in the Fibonacci sequence.

The problem of prime numbers has always been discussed in the mathematics community. The famous Riemann conjecture, Goldbach's conjecture, twin prime number conjecture, Mersenne prime number distribution, etc. are all problems related to prime numbers.

Although this kind of pure number game does not seem to be very useful, it has also attracted pure mathematicians to continue to explore.

The question of whether there are infinitely many prime numbers in the Fibonacci sequence discussed by Xu Hongbing is not as famous as the Mersenne prime number, but if the proof is completed, it will definitely cause a shock in the mathematics world, and it will also be of great use to him.

Like the classmate Liu who proved Sitapan’s conjecture, after graduating, he was directly employed as a full professor at Nanzhong University at the age of 22. So if Xu Hongbing can complete the proof, what kind of benefits he can get? It is unknown, especially the field of number theory to which this problem belongs, which is an area of ​​considerable concern to many mathematicians.

But obviously, he has been researching for many days, and he still has no clue. The wasted draft paper on the desk in front of him proves how much effort he has spent.

However, Xu Hongbing's problem now is not how to prove this problem, but, what about his problem?

After he looked for the paper on his desk, he found that the paper on which he wrote the Fibonacci sequence and asked if there were infinitely many prime numbers was missing.

Where did his question go?

Is it possible...

Xu Hongbing thought of a funny thing, wouldn't Lin Xiao think it was his question and take it away?

It's really possible.

Shaking his head, take it as a 'surprise' for Lin Xiao.

Oh no, it should be 'scared'.

Of course, the premise is that Lin Xiao can write there.

He has come up with eight difficult questions that he is very satisfied with. The first four questions can definitely become the final questions in the university's advanced algebra exam, and the last four questions are definitely competition-level difficult questions, and they are also the kind of mathematics professional group. Well, if Lin Xiao could finish all the writing before the self-study tomorrow night, he would just eat the desk in front of him.

With those scratch paper.

At that time, Lin Xiao didn't make it, so he came to him, and he just comforted Lin Xiao, it's okay, taught him that this showed the importance of having a teacher, and then persuaded him to study hard with the teacher after he went to college, and don't work behind closed doors .

I believe that Lin Xiao will also follow his guidance at that time, and the premise will definitely be more limitless in the future.

Thinking of his plan, Xu Hongbing nodded in satisfaction, we must be considered a good teacher.

Afterwards, without thinking too much, he found a new draft paper, wrote down the expression of the Fibonacci sequence, and then added another [Are there infinitely many prime numbers? 】

Then, he began to try to solve this problem again.

……

Hotel, in Lin Xiao's room.

"Well, there's something wrong with the teacher's question. It's really difficult."

Lin Xiao looked at the question Xu Hongbing gave him, and fell into thinking.

He had already finished writing the four questions on the first sheet of paper, and now he was writing on the second sheet of paper, but because the difficulty suddenly increased a lot, he also felt the difficulty for a while.

If he knew that this question could be called a question in the undergraduate mathematics professional competition, he would not be so entangled, because when he faced this kind of problem, he would need a certain amount of time to think about it. After all, the depth and breadth of university mathematics, There are also knowledge aspects that have certain requirements for students.

So, it took about 5 minutes, and he finally came up with an idea.

"It took me 5 minutes, it's ridiculous."

Shaking his head, he then began to write according to the ideas he had come up with.

That's right, for him, 5 minutes is considered a certain degree of difficulty, otherwise, like the question Xu Hongbing gave him during self-study just now, he could solve it in just 1 minute, more than three points species, all belong to partial difficulty.

"what are you doing?"

At this time, Kong Huaan, who was also studying on the other side, asked him.

But Lin Xiao didn't notice what he said because he was in a state of immersion.

Seeing that Lin Xiao didn't answer, Kong Huaan couldn't help but came to Lin Xiao curiously and took a look.

After reading it, he went back silently.

Sure enough, he should be a hacker himself, mathematics is not something he should consider.

Time passed quickly, and it was 11:30.

Lin Xiao, who had been working on the questions, finally heaved a sigh of relief and put down the second piece of paper.

Looking at the time, Lin Xiao couldn't help but feel that Xu Hongbing's ability to come up with questions is really great.

It took him an hour and a half to finish writing the eight questions, the first four questions only took him 10 minutes, and the last four questions took him 10 minutes.

In particular, these questions seem to be made by Xu Hongbing on the spur of the moment.

Being able to solve so many difficult problems casually is not something any teacher can do.

"But it's finally finished, as expected of me."

With a sigh in his heart, Lin Xiao picked up the third piece of paper.

The piece of paper with only one question on it.

I saw it read: [The first two items of the sequence Fn are both 1, let Fn3=Fn1+Fn2, may I ask whether there are infinitely many prime numbers in the sequence Fn? 】

Chapter 42 Difficulty

After reading the question, Lin Xiao's expression suddenly became serious.

This question is very difficult!

And it's not that hard.

Actually asked him to prove that there are infinitely many prime numbers in such a sequence?

It is easy for him to prove that there are infinite prime numbers in natural numbers, but it is not a simple matter to prove that there are infinite prime numbers in this sequence, because whether there are infinite prime numbers in a sequence can be called a kind of randomness. The event has happened, and it is quite difficult to complete it.

Lin Xiao couldn't help but fell into thinking.