Into Unscientific

"."

bench.

Looking at Avelyn who was humbly asking for advice, Xu Yun couldn't help but feel a little subtle.

well known.

Humans have three illusions:

someone is looking for me

I can fight back,

He/she likes me.

As a descendant who is very countable.

Although Xu Yun was not so narcissistic that the girl would confess to him, when he heard that the girl had a question to ask him, he more or less subconsciously thought that the other party would say something related to his background.

The result was unexpected.

Is the problem Avelyn mentioned really a problem?

Fibonacci sequence.

This is a very, very famous mathematical puzzle that is extremely useful in both mathematics and life and nature.

The Fibonacci sequence can be traced back to the 7th century AD, when there was a mathematician named Gopala in India.

This person first described this sequence when studying the number of methods when the lengths of the box packaging objects are exactly 1 and 2, which is the following problem:

There are n steps, and you can only cross one or two steps at a time. How many ways are there to go upstairs?

Then the question changed again and advanced to the more famous rabbit puzzle:

Assuming that rabbits have the ability to reproduce after two months of birth, a pair of rabbits can give birth to a pair of young rabbits every month.

If none of the rabbits died, how many pairs of rabbits could be bred in one year?

This problem was finally summarized by Fibonacci into a sequence, that is:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377... such an infinite sequence.

Its characteristic is that the latter number is the sum of the first two numbers, 0+1=1, 1+1=2, 1+2=3 and so on.

And dividing the next number by the previous number is infinitely close to the golden section number 0.618.

If this sequence is expressed in a formula, it is Xn=X(n-1)+X(n-2), where X0=0 and X1=1.

In the novel "The Da Vinci Code".

The curator of the Louvre Museum was killed and left dead on the floor. At that time, the curator took off his clothes, placed him in the Vitruvian Man, a famous painting by Leonardo da Vinci, and left some strange codes.

And these elusive passwords are the Fibonacci sequence.

The bee family tree, pinecone phyllotaxy, and even the shape of melons and fruits in nature are all related to the Fibonacci sequence. In 2005, Professor Cao Zexian cooperated with the Institute of Physics of the Chinese Academy of Sciences to study microbes with a diameter of about 10 microns using silver cores and silica shells. stress in the structure.

Finally, the Fibonacci spiral pattern was successfully generated by manipulating the stress on the inorganic microstructure composed of silver core and silica shell.

The more you study mathematics and physics, the more you will marvel at the wonder of life.

correct.

Now that Professor Cao Zexian is mentioned, here is a simple rumor by the way.

This Professor Cao Zexian is also a very controversial celebrity. He is the chief scientist of the 973 nanometer material project of the Ministry of Science and Technology, and a leader at the level of the Hundred Talents Program.

However, some outrageous views often come out of the mouth, some of which are true and some are false.

For example, he once said such a sentence in a lecture at the National University of Science and Technology:

"85% of the knowledge of mathematics and physics has not been introduced to China, and this knowledge is tightly held by foreigners."

This sentence is actually a bit bluffing, a bit deliberate and uttered wild words.

Everyone knows that there must be some knowledge abroad that has not been shared with us, but that content mainly covers the front-end field, and it is definitely not as outrageous as 85%.

So then.

Another sentence that was uttered with him at the time and used to support the above point of view has also become a joke on the Internet:

"You don't know, there are 44072 hearts in a triangle."

But in fact this sentence is true, and is a very formal direction of mathematical research.

It's just that it belongs to the conclusion of elementary plane geometry. Pingji has long been no longer the research direction of front-end mathematics, and it is basically not used by most people.

So it’s not that this knowledge has not been introduced into China, but it’s meaningless to teach it—even the top competition classes of top foreign universities will not conduct research on these triangle hearts.

Generally speaking.

Ordinary people only need to master the five minds, and those who learn geometry can master at most 50 kinds.

After that, it almost belongs to the category of pure theory, which is extremely unpopular and remote.

Therefore, Professor Cao used this example to prove that "85% of mathematics and physics knowledge has not been introduced to China" is not correct, but the number itself is not a problem.

It's not anti-intellectual, and it's not civil science, because the judgment of the triangle heart is that the three lines have the same point, so there are too many hearts locked.

There is currently a website that collects these hearts together at faculty.evansville.edu/ck6/encyclopedia/ETCPart4. (After all, this professor is a volute, and the content of the mouth is flat, but the data is indeed correct)

OK, back to the original topic.

The Fibonacci sequence is widely used in life and mathematics, and what are the perfect square terms in it has always been a very contradictory question.

so-called perfect squares.

Refers to the form in which a number can be expressed as the square of an integer.

For example, 4=2^2, 9=3^3, 256=4^4, etc.

Why is it said that the perfect square term in the Fibonacci sequence is a very contradictory question?

the reason is simple.

This problem was not calculated by the British mathematician J. H. E. Cohn until Xu Yun traveled more than 50 years ago, that is, in 1964.

In terms of time, it is undoubtedly a difficult problem that has only been solved in modern times.

But at the same time.

Its cracking process uses the content of elementary number theory, which is the same property as the prime number theorem and the four-color theorem.

This is also one of the very few mathematical problems that can be solved with elementary number theory. In theory, it could actually be solved in 1800.

Of course.

The very few examples in the past did not include Ge Guess - if you are lucky, you can see thousands of elementary proofs of Goldbach's conjectures born from domestic and foreign minkes every year.

But just like physics can be divided into classical physics and more microscopic quantum physics.

J. H. E. Cohn, that is, the complete square term proved by Cohen is only an answer within a certain range, and it is more recognized that the range of the first 200,000 Fibonacci numbers.

If the range is infinitely expanded, several more perfect square terms can still be found.

For example, the fourth number is 884358447525575649, which is about 1056412078.

Then there are 6.1613e+030, 9.9692e+030 and so on

This also belongs to the scope of theoretical research. For the current Aveline, it is enough to use Cohen's problem-solving method.

Then Xu Yun took the paper and pen, and began to calculate while talking:

"First of all, let's define a Lucas sequence, which is the Fibonacci sequence, Xn=X(n-1)+X(n-2), but X belongs to N, N≥3"

"Then extend the domain of definition from the set of natural numbers to the set of integers. We can get 2F_{m+n}=F_{m}L_{n}+F_{n}L_{m}"

"If m=1, we can get 2F_{n+1}=F_{1}L_{n}+F_{n}L_{1} and thus 2L_{m+n}=5F_{m}F_{n}+L_ {n}L_{m}”

"Then go in and out like this (mathematical induction method) to accelerate and decelerate (quadratic remainder). Then polish it a bit (Euler's discriminant method), touch it twice from this position (rolling and dividing method) and then nine shallow and one deep (modulo periodic sequence)"

After more than ten minutes.

".To sum up, 1,1,144 is the only perfect square term in the Fibonacci sequence!"

Xu Yun put down his pen, took a deep breath, and said to Aveline:

"Done!"

Avelyn took the arithmetic paper and looked at it carefully.

Xu Yun leaned back on the bench and wiped the sweat from his forehead in the blind spot of Aveline's vision.

It's finally done.

It should be OK next time, right?

However, just when Xu Yun thought he had passed the test, Avelyn's voice suddenly sounded in his ears:

"Student Luo Feng, when did you solve the problem of the perfect square term in the Fibonacci sequence?"

Xu Yun's mentality was relatively relaxed at this time, and he subconsciously opened his mouth when he heard the words:

"Nineteen"

But before he finished speaking, he suddenly came to his senses, and saw him sitting up straight quickly, and said with a dry smile:

"Student Avelyn, look at what you said, what problem did I solve?"

"This is the calculation result I discovered from the manuscript left by the ancestor of Fat Fish when I was nineteen years old."

Avelyn glanced at him with a half-smile, and confirmed:

"What you said is true?"

Xu Yun had a faint feeling of bad premonition, but now that the words have been spoken, there is no reason to take them back:

"Of course it's true. I'm a sincere young gentleman who claims to change 30,000 a day."

Avelyn watched him quietly for a few seconds, then suddenly took out two manuscripts from her body and handed them to Xu Yun:

"Then look at this."

Xu Yun subconsciously took the manuscript, put it in front of him, and began to read it.

The first manuscript seems to be a bit old, the handwriting is messy, and it has a taste of letting go, but there is an inexplicable sense of familiarity.

The handwriting of the second manuscript was much more delicate and neat, and Xu Yun recognized it as Avelyn's handwriting at a glance:

On Christmas Day, everyone wrote down their future expectations in the diary, and Xu Yun still remembers Avelyn's handwriting and content.

In addition to the difference in handwriting between the two manuscripts, the contents above made Xu Yun's eyes widen:

Although the problem solving methods are different, they are all demonstrating the perfect square term in the Fibonacci sequence!

The method of the first manuscript is relatively primitive, and the starting point is Fermat's little theorem.

Then it was transformed through Taylor's formula of n times unit root, and "self-study" produced a relatively primitive odd prime number check logic.

Avelyn's derivation process is relatively simple in terms of tools, but the steps are slightly cumbersome.

Her process can be simplified in some places, but the main idea is the same as that of Xu Yun

Totally agree!

no doubt.

Long before Xu Yun spoke, Avelyn had mastered at least two methods of solving the problem.

Seeing that Xu Yun was swallowing saliva, Avelyn continued to add the knife:

"Student Luo Feng, as you can see, the first manuscript is the derivation process left by Newton's ancestors, and the second manuscript is my bad work."

"Euler was less than 20 years old when Newton's ancestor was alive, and he was far from deriving Euler's discriminant method."

"So although he solved the problem in the Fibonacci sequence, he only used a logical tool created by himself, and other ideas were relatively primitive."

"At the same time, Ancestor Newton and Mr. Fat Yu are also teachers and friends. They love to compete with Mr. Fat Yu in everything, so he once left a sentence after calculating this result."

Speaking of which, Avelyn looked up at Xu Yun, and said:

"He said, 'If Fat Fish can solve this problem, the only way is to develop a logical tool through Han Li's self-study like me.'"

"However, in your calculation process, Euler's discriminant method has been widely used. This is the method that Euler induced in 1757."

"."

Xu Yun was silent for a few seconds, feeling that he should save himself again:

"Student Avelyn, couldn't it be that the ancestor of Fat Fish deduced this rule before Ola?"

Avelyn shook her head, took out an older manuscript from her body, and said:

"Ancestor Newton once encountered a huge bottleneck when calculating the infinite magnitude. At that time, Mr. Fat Fish once proposed a quadratic approximation formula, which is this."

Xu Yun was slightly taken aback, then took the manuscript paper.

There is not much content on the paper, only a formula is listed:

V(r)≈[V''(re)/2!](r-re)^2. (Chapter 32, close the foreshadowing, buried 1.5 million words, let me have a hip for a while, but it’s awesome)

Upon seeing this, Evelyn added:

"From this formula, it can be seen that Mr. Fat Yu's thinking does not follow the law of quadratic reciprocity, and it is completely different from Euler's system."

"You should know that for a mathematician, the thinking system is not something that can be easily transformed."

After speaking, she took back her manuscript from Xu Yun's hand, and shook it in front of Xu Yun:

"In addition, your derivation process and mine are almost the same. The whole process has an obvious post-Euler color, and it is absolutely impossible to be the result of a hundred years ago."

"so."

Avelyn's eyes were as bright as jewels in the warm sun, and her ethereal voice hit Xu Yun's heart directly:

"Including some of the previous experimental designs, quite a few are actually from your own hands, am I right?"

"."

Xu Yun was silent.

be honest.

Ever since Avelyn discovered the loophole in the name of the photovoltaic effect, he has actually been avoiding another rollover.

For example, the relativity equation he gave Gauss, and various links in the cathode ray, etc., have undergone a lot of magical changes

but the problem is

Most of the content he involved in the experimental session was related to physics.

But this time Aveline proposed a math problem.

You should know that most of the physical knowledge can be divided into stages.

for example.

The previously mentioned Lorentz force formula f=qVBsinθ.

Before this formula was generalized in 1895, unless you were a traveler, it was impossible to calculate the Lorentz force under such and such conditions.

But mathematics is different.

Many concepts in mathematics are incremental.

That is, before a certain formula is summarized, you actually have a certain chance to find its prototype.

For example, how much work A has completed in a certain interval, B has supplemented it after him, and finally C spreads this law to a larger range-such as an integer set and so on.

So at least for a physicist like Xu Yun.

You ask him to consider whether Euler's judgment has been established when solving elementary number theory. This is actually a very difficult detailed problem.

High mathematical sensitivity is required.

If he has enough time to think or distinguish, that's better, maybe there is a higher probability of a patch or something.

But Avelyn appeared too suddenly today, and the initiative of the topic was not in Xu Yun's hands.

Therefore, the successive factors coincided, and Xu Yun made another huge, huge, super super super mistake this time:

He used the derivation system of Euler's discriminant method, which is the related method he learned later.

So he was possessed by Xiao Heizi, showing his chicken feet.

Looking at Avelyn with a determined face in front of her, Xu Yun couldn't help but toss the "Classical Physics" in her hand.

If she denied it, wouldn't this girl let herself feel the power of knowledge?

Besides, as far as the current situation is concerned, it doesn't make much difference whether you deny it or not.

Think here.

Xu Yun couldn't help but sighed faintly, and nodded very bachelorly:

"Um."

Hear this answer.

A smile suddenly appeared on Evelyn's face.

The curvature of the corner of the mouth is as perfect as a crescent moon, like a ripple on the face, quickly across the face:

"It seems... I guessed right, you are actually a genius, a real genius, right?"