From Almighty Scholar to Chief Scientist

Chapter 42 Difficulties
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.

Mr. Xu should give him advanced algebra questions, right?
But this question doesn't look like a question in the direction of advanced algebra?

Obviously it is a number topic, and of course number theory can also be solved with knowledge of algebra.

So is polynomial?

matrix?

Or spatial or linear functions?
The questions the teacher gave him couldn't be unsolved math problems, could they?

It is definitely possible to solve it, but it is a bit difficult...

So, he thought hard for 5 minutes like this, and at the same time performed a simple calculation on the draft paper.

In calculus, we must first list the laws of this sequence of numbers.

Lin Xiao listed the first few items of the sequence.

1, 1, 2, 3, 5, 8, 13, ...

Seeing these series of numbers, he was taken aback for a moment. This series of numbers seemed familiar. After a quick thought, isn't this the Fibonacci series?
No wonder, when he looked at this general term formula, he felt a little familiar.

The Fibonacci sequence, named after the Italian mathematician Leonardo Fibonacci in the twelfth century, is defined recursively in mathematics: the zeroth and first terms are specified After being 0 and 1 respectively, each of the remaining items is equal to the sum of the first two items, and the zeroth item is a special item and is not included in the sequence.

You may think that this sequence looks ordinary, isn't it such a simple law, I can also create a sequence.

For example, it is called the Zhangsan/Outlaw Fanatics sequence, which stipulates that the first three items are 1, and each of the remaining items is equal to the sum of the first three items, or the first four items are stipulated.

However, the reason why the Fibonacci sequence is special is that it is not so simple. The Fibonacci sequence is also called the golden section sequence. The value of the previous item divided by the next item will get closer and closer Based on the golden ratio, which is 0.618.

In addition, there are many coincidences in this sequence in nature. For example, 99% of the spiral arrangement of sunflower seeds obeys the Fibonacci sequence, and the growth law of branches also conforms to this sequence.

Therefore, there are many mathematicians who study the Fibonacci sequence.

However, this Fibonacci prime problem...

Lin Xiao was confused.

Isn't this really an unsolved problem in mathematics?
But this is a question the teacher gave me...

It's impossible for Mr. Xu to cheat him on purpose, right?
Or, he took the wrong question?

Why don't you search your phone?
But after thinking about it, if this question has been solved, wouldn't he be considered to know the answer in advance?
For him, even seeing an idea is of great help to solving problems.

Lin Xiao didn't know that this was indeed an unsolved problem, because he didn't study the Fibonacci sequence, and he knew that the formulas of the general terms of this sequence were all calculated, so how could he understand these side details?
Moreover, this problem is not well-known. The unsolved mathematical problems generally known to middle school students in Huaguo are basically limited to Goldbach's conjecture, because there is a mathematician named Chen in Huaguo who solved Goldbach's conjecture. The "1+2" ​​problem, so for the purpose of propaganda, this problem was written in the mathematics textbook and told to the elementary and middle school students in Huaguo.

As for the more famous problems in mathematics, such as Riemann conjecture, BSD conjecture, Hodge conjecture, etc., not many primary and middle school students know it.

So Lin Xiao got tangled up and didn't know how to deal with this question.

But suddenly, a light flashed in his mind.

This question is written on the third piece of paper!
And the questions on the first paper are obviously easier than the questions on the second paper. From this point of view, the questions on the third paper must be more difficult than the second paper.

But the question on the second sheet was already difficult enough, and there was only one question on the third sheet, which was even more difficult, obviously it should be taken for granted.

This logic is easy to figure out!

Lin Xiao immediately stopped being entangled, and at the same time respected Teacher Xu Hongbing.

This kind of control over the difficulty of various topics before and after is really amazing!

As expected of a professor of mathematics.

So he stopped thinking too much and continued to think about his ideas.

In this way, 1 minute passed, 2 minutes passed, and 10 minutes passed.

Endless storms had already set off in his mind, and the synapses at the nerve endings released transmitters at a high frequency, causing his brain to start operating at an extremely deep level.

Soon, he had a flash of inspiration, if it is a polynomial...

He immediately started writing on the draft paper.

First write its general term formula as An-(An-1)-(An-2)=0.

"Then you can use the method of solving the second-order linear homogeneous recursive relation, then its characteristic polynomial is..."

[The characteristic polynomial is: λ-λ-1=0]

【得λ1=1/2（1+√5），λ2=1/2(1-√5)】

【即有An=c1λ1^n+c2λ2^n，其中c1，c2为常数，我们知道A0=0，A1=1，因此……】

【最终解得c1=1/√5，c2=-1/√5。】

[Introduce the prime number theorem here, π(x)= Li(x)+ O(xe^(-c√lnx)(x→∞), where Li(x)=……]

After writing this, Lin Xiao fell into thinking again.

Next, he will try to combine the two.

As long as the two can be combined, then he has completed the proof.

Because, the prime number theorem is obviously based on the conclusion that there are infinitely many prime numbers, as long as the two can be contained, and the area is infinite, then the conclusion can be drawn.

That is to say, if one is proved to be large, then the small one will naturally complete the proof.

But obviously, it is not easy to combine the two and find the connection point, and more processing is needed in the middle.

"They need to be changed, the relationship between the two is too far away now..."

Lin Xiao stroked his chin, thinking about how to transform them equivalently.

At this moment, he felt a pat on his shoulder.

"Lin Xiao? Lin Xiao?"

He recovered and looked to the side.

It's Kong Huaan.

"what happened?"

Lin Xiao asked.

"It's almost twelve o'clock, don't you rest yet?"

"Huh? Is it twelve o'clock?"

Lin Xiao realized that it was very late, even if he didn't take a rest, Kong Huaan still had to.

So he could only temporarily give up and continue thinking, nodded and said: "Well, I'm going to rest."

Then he closed the draft paper and went to wash. After washing and returning to the bed, he was still thinking about how to prove it next.

Gradually, however, he fell asleep.

No way, he fell asleep on the bed.

(End of this chapter)