After reading the question, Lin Xiao's expression suddenly became serious.
This question is difficult!
And it's not that difficult.
He was actually asked to prove that there are infinitely many prime numbers in such a sequence?
It's easy to ask him to prove that there are infinite prime numbers in the natural numbers, but proving that there are infinite prime numbers in this sequence is not a simple matter, because whether there are infinite prime numbers in a sequence can almost be called a kind of randomness.
It's an event, and it's quite difficult to complete it.
Lin Xiao couldn't help but fell into thinking.
Teacher Xu probably gave him advanced algebra questions, right?
But this question doesn’t look like it’s a question in the direction of advanced algebra?
It is obviously a number problem, and of course number theory can also be solved using algebraic knowledge.
So it's a polynomial?
matrix?
Or space or linear function?
The questions the teacher gave him couldn't be some unsolved mathematical problems, right?
It can definitely be solved, it’s just a little difficult...
So, he thought hard for five minutes and performed simple calculations on the scratch paper.
To perform calculations, we must first list the rules of this sequence.
Lin Xiao lists the first few items of the sequence.
1,1,2,3,5,8,13,…
Seeing these sequence of numbers, he was suddenly startled. This sequence seemed familiar. He quickly thought, isn't this the Fibonacci sequence?
No wonder, when he saw this general formula, he felt that it looked familiar.
The Fibonacci sequence, named after the twelfth-century Italian mathematician Leonardo Fibonacci, is defined recursively in mathematics: specifying the zeroth and first terms.
After being 0 and 1 respectively, the remaining items are 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, but it is such a simple rule. I can also create a sequence.
For example, it is called Zhang San/Outlaw Sequence, which stipulates that the first three items are 1, and each remaining item is equal to the sum of the first three items, or the first four items are specified.
However, the reason why the Fibonacci sequence is special is that it is not that simple. The Fibonacci sequence is also called the golden section sequence. The value of the previous term divided by the following term will become closer and closer.
In the golden ratio, that is 0.618.
In addition, there are many coincidences in this sequence in nature. For example, 99% of the spiral arrangement of sunflower seeds follows the Fibonacci sequence, and the growth pattern of branches also conforms to this sequence.
Therefore, there are many mathematicians who study the Fibonacci sequence.
However, this Fibonacci prime number 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 Teacher Xu to deliberately trick him, right?
In other words, he took the wrong question?
Why don't you take your phone and search it?
But after thinking about it, if this question has been solved, doesn't he know the answer in advance?
For him, even seeing an idea is of great help in solving problems.
Lin Xiao didn't know that this was indeed an unsolved problem, because he didn't study the Fibonacci sequence. If he knew the general formula of this sequence, how could he understand these details?
And this problem is not famous. The unsolved mathematical problems that Chinese middle school students generally know are basically limited to Goldbach's conjecture, because a Chinese mathematician named Chen solved the problem in Goldbach's conjecture.
The "1+2" problem was written in mathematics textbooks for publicity purposes and told to primary and secondary school students in China.
As for the more famous problems in mathematics, such as Riemann's conjecture, BSD conjecture, Hodge's conjecture, etc., not many primary and middle school students know about them.
So Lin Xiao became entangled and didn't know how to deal with this problem.
But suddenly, an idea flashed in his mind.
This question is written on the third piece of paper!
The questions on the first sheet of paper are obviously easier than those on the second sheet. From this point of view, the questions on the third sheet of paper must also be more difficult than those on the second sheet of paper.
The question on the second sheet of paper is already difficult enough, but there is only one question on this third sheet of paper, which is even more difficult, so obviously it should be taken for granted.
This logic is easy to figure out!
Lin Xiao immediately stopped worrying, and at the same time, he was in awe of Teacher Xu Hongbing.
This kind of control over the difficulty of various questions before and after is really amazing!
He is worthy of being a professor of mathematics.
So he stopped thinking too much and continued to think about his ideas.
Just like that, one minute passed, two minutes passed, and ten minutes passed.
An endless storm had set off in his mind, and the synapses of his nerve endings released transmitters at a high frequency, causing his brain to start operating at a very deep level.
Soon, he had an idea. If it was a polynomial...
He immediately started writing on the scratch paper.
First, write its general formula as An-(An-1)-(An-2)=0.
"Then we can use the method of solving the second-order linear homogeneous recursive relationship, then its characteristic polynomial is..."
[Characteristic polynomial is: λ-λ-1=0]
[Get λ1=1/2(1+√5), λ2=1/2(1-√5)]
[That is, An=c1λ1^n+c2λ2^n, where c1 and c2 are constants. We know that A0=0, A1=1, so...]
[The final solution is c1=1/√5, c2=-1/√5.]
[The prime number theorem is introduced here, π(x)= Li(x)+ O(xe^(-c√lnx)(x→∞), where Li(x)=...]
Writing this, Lin Xiao once again fell into thinking.
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 included and the areas are both infinite, the conclusion can be drawn.
That is, if you prove a big one, the small one will naturally complete the proof.
But obviously, it is not easy to combine the two and find the connection points, and more processing is required in the middle.
"They need to be changed. The relationship between the two is too far now..."
Lin Xiao rubbed his chin, thinking about how to equivalently deform them.
At this moment, he felt a tap on his shoulder.
"Lin Xiao? Lin Xiao?"
He came back to his senses and looked to his side.
It's Kong Hua'an.
"What's wrong?"
Lin Xiao asked.
"It's almost twelve o'clock, why don't you take a rest?"
"Ah? Is it already twelve o'clock?"
Lin Xiao realized that it was already very late. Even if he didn't rest, Kong Huaan also had to rest.
So he could only give up and continue thinking for the time being, nodded and said: "Well, get ready to rest."
Then he closed the scratch paper and went to wash up. After washing up and returning to bed, he was still thinking about how to prove it next.