Looking at the brand new manuscript that Gauss handed to him, Xu Yun could not help but feel a sense of curiosity on his face.
What could be the content here?
To know.
In the field of mathematics, affinity numbers belong to a branch of number theory.
If you really want to count the "relatives" it can match, there are too many examples that meet the criteria.
For example, prime numbers, equal sums, isolated numbers, common sums, etc. are all...
Even if you insist on talking about it.
Non-Euclidean geometry can all be related to number theory:
Because non-Euclidean geometry is also a formal system of first-order predicate logic and elementary number theory, which conforms to Gödel's incompleteness theorem.
Therefore, Xu Yun really couldn't guess the content of this manuscript based on Gauss's introduction alone. He could only know it by reading it in person.
Then he stretched out his hands and carefully took the manuscript.
Then he thought of something again, stopped his movements, and asked Gauss:
"Professor Gauss, you gave me this manuscript. I'll finish it after reading it..."
As a result, before Xu Yun finished speaking, Gauss ruthlessly dismissed his idea:
"Of course it must be recorded in one of the five volumes."
Xu Yun could only shrug.
Well, the card logic bug failed.
But overall it's not a big problem, after all, the opportunity of these five volumes of manuscripts is an unexpected surprise.
Then he looked at the outside of the manuscript and found that the manuscript was only tied with a red ribbon, and there was no seal with rough content like affinity numbers.
See this situation.
Xu Yun's eyes suddenly condensed, and the importance in his heart increased a bit:
The fact that the manuscript can be found without indexing the title shows that its status in Gauss's mind must be extraordinary, at least without the need to rely on a seal as a memory reminder.
Think of this.
Xu Yun couldn't help but move a little faster to untie the ribbon. It looked like he was untying...his shoelaces.
Well, untie your shoelaces and don't think too much about it.
A little less than half a minute later.
A flat roll of manuscript paper appeared in front of Xu Yun.
Xu Yun held the two corners of the upper half of the manuscript paper, picked it up as if he was holding the author upside down, and read it line by line.
A few seconds later.
Xu Yun's pupils shrank suddenly, and in shock, the manuscript in his hand almost fell from his hands!
I saw a line of words written at the beginning of this manuscript:
"Proof of the Existence of Odd and Perfect Numbers"
The correct pronunciation of this title is [Proof of the nonexistence/of odd perfect numbers], and the most critical core is the two words in the middle:
Odd perfect numbers do not exist.
Students who understand number theory should all know this.
If these two words appear at the same time in 2022, they are destined to cause a big earthquake in the mathematics world.
Mentioned earlier.
In 2022, when Xu Yun traveled back in time, the status of affinity numbers in the mathematical world has always been somewhat awkward:
on the one hand.
The affinity numbers can be listed exhaustively by computer, and the sum of the divisors can be compared like a production line.
If the conditions are met, the output is YES, otherwise it is NO, and it can be done with one click.
As of 3:34 a.m. on August 15, 2022, the number of affinities that have been discovered exceeds the number of pairs.
The longest pair of numbers is more than 24 million digits long - please note that it is not the number 24 million, but 24 million digits, and one hundred million is nine digits.
If you really don't understand this concept very well, you can think of "bit" as a word.
24 million digits, which is equivalent to an online novel of 24 million words.
If the author lists this number, the number of words in our book will immediately jump to the first few...
In fact, this is not the most outrageous, the mentioned pi is the most scary - it has been calculated to 100 trillion digits. (Thanks to the reader for correcting me, I checked and the 62 trillion record has indeed been refreshed, it is only eight
Less than a month, too fast)
The person who set this record was Google Cloud engineer Emma Haruka Iwao, a neon person.
He used 25 Google virtual machines, which took 158 days, and finally set this record in June this year.
This is also the project leader who calculated 31.4 trillion digits of pi in 2019, but compared to his achievements, his orientation is also quite subtle:
From the previous words, it is not difficult to see that this boss is a lesbian supporter who is biologically female but psychologically male...
So Xu Yun sometimes wonders, do capable people these days like to add buffs to themselves?
OK, let’s return to the original topic.
Since the computer can filter out affinity numbers with so many digits, why do you still call it embarrassing?
the reason is simple.
That is, the specific laws of affinity numbers have not been completely cracked, and computers only rely on the exhaustive method.
This method leads to the emergence of another part of these affinity numbers that are 'mutated' and unknown.
For example.
If you add up its divisors, you get this number.
Then add up the divisors of , you will get;
Then continue this process.
will become...
will become...
It will become...
That's right.
After five changes, I'm right back.
This kind of number is called a communicative number.
Because its circle of friends is wider than the affinity number...or the blind date number, some people also call it the Neptune number.
In addition to the communicative numbers, there is another number that is equally special to the extreme.
That is a perfect number, also called a perfect number.
The concept of this number is actually very simple:
When you add their divisors, you get themselves.
The smallest example is 6.
The divisors of 6 are 1, 2 and 3, and 1 2 3=6.
Next is 28, because 28=1 2 4 7 14.
The next perfect number of 28 is 496, and then there is a larger leap to 8128.
As for what happens next...
It's getting more and more ridiculous.
For example, the next perfect number of 8128 is, followed by, and followed closely by.
The person behind me is the one who is holding me back. It looks like he is applying for an ID card...
As of the time Xu Yun traveled through time, there were only 51 complete ones in total.
The currently largest known perfect number was discovered in 2018. It has digits and as many as several divisors.
It is equivalent to a novel with 49 million words, which is fully twice the maximum affinity number above. Adding the two together, only "The Flash Student at the Giant School" on the entire Internet has more words than it...
This is actually a very nerve-wracking thing:
Think about it.
Its divisors, the sum of which is exactly equal to itself...
Therefore, the reason why many people in later generations believe that mathematics hides the mysteries of the universe is not because they are making flattering remarks to increase the attention of their own industry, but because some numbers are truly exquisite to the extreme.
In addition, the subject of mathematics also reflects the dark and cruel reality of the universe from a philosophical perspective - if you don't know it, you just don't know it. You can only get one point for writing a solution, and even gods can't save you...
Ahem...
In addition to the divisor properties, perfect numbers have two special features:
One is that all perfect numbers discovered so far correspond to Mersenne primes one-to-one, without exception.
In other words, there are as many perfect numbers as there are Mersenne primes found.
Today, a project team called GIMPS is performing relevant calculations. In 14 years, a total of 10 Mersenne prime numbers...or perfect numbers have been found.
The Chinese national team currently ranks eighth in terms of contribution to this project group, with a total contribution of about 1.5%.
By the way, I would like to share a website called equn., which is the official website of China Distributed Computing Station.
If you want to make a small contribution to the research of mathematics or other natural sciences in your own way, you can choose a project that suits your taste and apply to join.
Except that the perfect numbers all correspond to Mersenne primes one-to-one.
The second special thing about perfect numbers is that...
All perfect numbers discovered so far are even numbers, ending in 6 and 28.
Later generations have not found an odd perfect number, but there is also no proof of its non-existence.
The only knowledge about odd and perfect numbers in 2022 is the proof proposed by Austin Ohr:
If there is an odd perfect number, its form must be in the form of 12^p 1 or 36^p 9, where p is a prime number.
That is to say, even if there is an odd perfect number, it must be at least 10 to the 1500th power.
Then it was gone.
That's right, no more - there is basically no progress in the theoretical direction for odd and perfect numbers in the mathematical community.
Of course.
This means that no results have been produced, but it does not mean that everyone has given up on relevant computing work.
But what Xu Yun didn't expect was...
This problem that has caused headaches and even baldness for countless people in later generations, Gauss seems... seems... probably... maybe... seemingly...
Was it settled in 1850?
Oh my god!
Xu Yun dared to bet that his own manuscript did not exist at all. There must not be such a manuscript among the "relics" of Gauss in later generations!
Think of this.
Xu Yun could no longer restrain his excitement and began to read it carefully.
The first volume of the manuscript is not a calculation and derivation process, but a diary-like essay.
"1831 Alley, September clear, Faraday updated Chapter 7, the generator continues to push the next line of human development..."
"On September 15, after attending Mina's funeral, I felt extremely sad."
"After seven days of silence, Therese's reciting voice suddenly came from the window, [Mr. Fat Fish helped young Sir Newton and said to him, Mr. Newton, the car is ready, don't stop]!"
"The words of the sages are like the light in the dark night, giving me the courage to look forward again."
“It happened that Dirichlet was visiting, and I saw the ‘Unsolved Mysteries of Mathematics’ revised by the University of Würzburg in his hand, and I started to become more playful.”
"So I wrote down a few small pieces of paper, folded them into a ball, and asked Therese to randomly pick one of them. The question on it was 'Do odd perfect numbers exist?'"
"Then I spent four hours and thirty-five minutes writing this, pulling up my pants, and reviewing... the general stuff."
Xu Yun:
"...."
Then he took a deep breath and turned to the next page.
As soon as he turned the page, a large and obvious word appeared in front of him:
untie.
untie:
"Everyone knows."
"A positive integer n is an even perfect number if and only if n=2m?1(2m?1)n=2^{m-1}(2^{m}-1)n=2m?1(2m?1
) where m, 2 m?1m, 2^{m}-1m, 2^m?1 are all prime numbers."
"Suppose p is a prime number and a is a positive integer, then we have:"
"σ(pa)=1 p p2 ... p^a={p^(a 1)?1}/p-1."
"Suppose a positive integer n has prime factorization n=p^(a1/1)p^(a2/2)p^(a3/3)....p^(as/s)."
"Since the factor and function σ are multiplicative functions, then:"
"In square numbers, the sum of their consecutive additions is multiplied by 6. Some are evenly divisible by n times n plus 1, which is equal to 2n plus 1. That is, 2n minus 1 is a prime number, and 2n plus 1 is a prime number, so it is a pair of twins.
Prime number."
"In the continuous addition of 2nd power and 5th power, there is a form of 2 times 3 times 5 times 7... In mathematical calculations, conversely, it is to calculate the sum of continuous additions, and 1st power, 2nd power
If the powers are the same, write down the calculation form, that is, even numbers plus 1 and minus 1, which can be written as prime numbers and composite numbers..."
"That is, σ(n)≠2n, where n is an odd number greater than 1, and σ(1)=1, σ(1)=1."
"so......"
"There are no odd and perfect numbers." (Actually, the last step is impossible to pass. It's a trick, don't go into it too deeply. For inspiration, refer to 10.3969/j.issn.1009-4822.2009.02.003)
Look at the last sentence at the end of the pen.
Xu Yun was silent for a long time.
Thousands of words in my heart finally turned into a long sigh.
This is Gauss...
A man who stands at the pinnacle of the history of mathematics throughout the ages, a German who has conquered a wider territory than a certain mustache.
Xu Yun was mesmerized by this seemingly essay-like manuscript...
suddenly.
Xu Yun thought of what Gauss had said to him before:
"I don't create miracles because I am a miracle."
This short old man, with his talent and intelligence, suddenly became one of the highest peaks in the history of mathematics.
Even in the future generations that Xu Yun traveled through, there is still no one who can match him.
anyway.
Mavericks, Lao Su, Lao Jia, Faraday, plus today's Gauss...
Xu Yun could no longer remember how many times he had admired the wisdom of the sages.
If given the chance, I really want to write a novel about my experience...
And just when Xu Yun was in a state of mind.
Gauss's voice suddenly sounded in his ears:
"Classmate Luo Feng, how is the quality of this manuscript?"
Xu Yun then brought his thoughts back to reality, pondered for a moment, and said to Gauss seriously:
"Professor Gauss, in my opinion, this manuscript alone is worth ten pieces of piezoelectric ceramic preparation technology."
"Perhaps hundreds of years from now, science and technology will have developed to an extremely astonishing level, and humans will be able to fly all over the world and enter the earth, but they will still be amazed by your wisdom."
Xu Yun's words did not contain any exaggeration, because he really thought so.
The discoverers of the piezoelectric effect were the Curie brothers. To be honest, this technology can only be considered quite satisfactory.
There are many technologies that can replace piezoelectric ceramics in the future, but piezoelectric ceramics have the lowest cost, the most mature technology, and the difficulty of preparation is relatively simple.
But the odd-and-perfect number of manuscripts is different.
It is a problem that has troubled the mathematical community for nearly 350 years!
Although its status in later generations is not as good as the Riemann Hypothesis or Hodge Hypothesis, it is still a very important research direction.
Although no results have been published, this is not because no one studies it, but because it is too difficult...
Just like the lithography machine that many people are thinking about, you can say that there has been no successful breakthrough in the country, but you cannot deny that the country has not invested a lot of energy and financial resources in it.
So in Xu Yun's opinion.
A manuscript that can solve the problem of odd perfect numbers is indeed worth as much as the preparation process of ten piezoelectric ceramics.
And opposite him.
Seeing Xu Yun, the 'descendant of fat fish', praising himself so much, an uncontrollable smile suddenly appeared on Gauss's face - based on his life experience, he could naturally tell whether Xu Yun's exaggeration was true or false.
I saw him waving his hands with a "humble" look on his face, smiling and saying to Xu Yun:
"Classmate Luo Feng, I'm over-praising it. It's just a relatively ordinary achievement, not of that high value. By the way, can you repeat what you said above when Michael is here?"
Xu Yun: "......?"
Then he solemnly put the manuscript away again and placed it next to the manuscripts of Affinity and Numbers.
Then Xu Yun was about to look for the next volume of manuscripts, but when he was about to start, a flash of inspiration suddenly flashed in his mind.
He loves to eat watermelon, but he doesn't know how to choose. He is a vegetable and loves to play.
So every time he goes to the supermarket, he likes to ask those aunties for help.
Most aunts will help with a little effort when they are in a good mood.
Although Occasionally the aunt will overturn because of her poor skills, but most of the time the melons she picks out are much better than what he picked by himself.
And isn't the current process of selecting manuscripts just like selecting watermelons...
And this person is far more than just an aunt who goes to the market, she is a melon farmer who grows watermelons!
Which manuscript is helpful? Gauss must know better than Xu Yun!
Think of this.
Xu Yun quickly turned his head and looked at Gauss expectantly, his meaning was obvious:
Boss, can you help me pick another volume?
This chapter is not over yet, please click on the next page to continue reading! Gauss immediately understood Xu Yun’s thoughts, hesitated for a moment, shook his head and said:
"Classmate Luo Feng, I am making an exception by giving you five volumes of manuscripts. You still want me to go and select them myself. Isn't this a bit overreaching?"
"I won't provide any further advice. It's up to you what manuscript you choose."
Looking at the resolute Gauss, Xu Yun thought for a while and said:
"Professor Gauss, won't Mr. Faraday have a press conference for his new work in a few days? School leaders such as Mr. William Whewell will also appear. At that time, I can take advantage of the media's presence to praise your manuscript and Fat Fish.
There is no distinction between ancestors and uncles..."
Xu Yun didn't finish his words.
There was a flash before his eyes, leaving only an afterimage and Gauss's voice in the air:
"You stand here and don't move around. I'll pick out some manuscripts for you!"
Xu Yun:
"..."
Boss, you should be more reserved...
After arriving at the suitcase.
Gauss leaned down slightly and kept scanning the suitcase.
Which books should I choose...
A few seconds passed.
Suddenly his eyes lit up, he pulled out two thicker volumes of manuscripts, dusted off the non-existent dust, and handed them to Xu Yun:
"Classmate Luo Feng, if nothing else, you should be interested in these two volumes of manuscripts."
Xu Yun still took it with both hands and checked the external situation.
The two volumes of manuscripts have the same affinity numbers as the first volume, and both have related labels:
"Research on Superposed Light Fields"
"Operator Problems of Flow Metrics"
Then Xu Yun took them to the desk, spread them out as usual, and read them carefully.
For a later generation like Xu Yun, neither book is difficult.
For example, "Research on Superposed Light Fields" records the study of Fresnel diffraction by Gaussian, with some additional topological charge and azimuth angle data.
If someone studies in this direction, they will make some achievements in optical fiber output transmission.
"Operator Problem of Flow Metric" is a bit more complicated.
It involves the prototype of non-Euclidean geometry and Riemannian geometry, and adapts to the ordinary derivative operator of the Cartesian system?.
The difficulty of getting started is much higher than that of "Research on Superposed Light Fields". It can be said to be the pioneering achievement of Minkowski space and Rayleigh approximation.
Now Rayleigh is only eight years old, and Minkowski is even younger than 14 years old.
It is really amazing that Gauss was able to study to this extent one step ahead of them.
In addition, this manuscript also determines the order of the tensor. After Gauss's death, this manuscript will definitely bring great inspiration to Riemann's work.
But I admire it.
At this time, the fluctuation in Xu Yun's heart was not as great as when he saw the second volume of the manuscript.
because......
Be it "Research on Superposed Light Fields" or "Operator Problems of Flow Metrics".
The quality of these two manuscripts is obviously beyond doubt, but they have not been lost in later generations, and they are also among the few manuscripts of Gauss that have been thoroughly studied.
In this situation.
Xu Yun could not reach the level of 'ecstatic' under any circumstances.
Of course.
It cannot be said that Gauss slighted Xu Yun.
On the contrary, these two volumes of manuscripts are actually very valuable.
If they had appeared in 1850, I am afraid they would have caused greater repercussions than the odd perfect numbers - especially the latter, which was the prototype of fluid geometry.
The reason for the unequal ideas between Xu Yun and Gauss is not the quality of the manuscripts, but the differences in the eras in which they lived.
The completeness of the knowledge theory of the era caused the two to view issues on different levels.
However, the regret in his heart turned into regret, and Xu Yun did not show any other complex expression.
I am still very grateful to have accepted these two volumes of manuscripts.
After all, this was Gauss's intention. For Gauss today, these two volumes of manuscripts can be regarded as half-baked achievements.
Of the five volumes of manuscripts, four have now been selected.
Only the last volume is left undecided.
In this last volume, Xu Yun still asked Gauss to make the choice.
"The last volume..."
Gauss stood next to the suitcase, his eyes quickly scanning the suitcase.
Which manuscript should I choose for Xu Yun?
He had already given the core manuscript of non-Euclidean geometry to Wheat, and given the relationship between Wheat and Xu Yun, Xu Yun would definitely be able to see that manuscript.
Therefore, manuscripts related to non-Euclidean geometry can be excluded...
Should I choose the period calculation of the lemniscate function?
Or astronomical observations?
How about choosing the geometric representation of the quadratic model function that I completed last year?
None of it seems appropriate...
A few seconds passed.
Gauss suddenly thought of something.
By the way, that thing!
I saw him bend down and slowly pick up a letter that was placed independently on a certain mezzanine.
Then Gauss put the letter in his palm, and his old fingers slowly passed over the envelope, with a hesitant expression in his eyes.
Xu Yun noticed.
Gauss's look was not one of reluctance, but rather...
sad?
Xu Yun rubbed his eyes, wondering if he had seen it wrong - why did Gauss have such an expression on his face?
After two full minutes.
Gauss sighed, handed the letter to Xu Yun with a complex expression, and said:
"Classmate Luo Feng, if nothing else happens, the previous four volumes of manuscripts should be enough for you to study for a long time."
"So the last volume of manuscript I selected for you is not some intellectual achievement that has not yet been made public, but this..."
"letter."
........
Note:
Yesterday it was 7600 words, and today it is 8400 words. I have been coding all night until now. Isn’t it too much to ask for a monthly pass?
