Chapter 307 Gauss’s Treasure (Part 2) (7.6K)
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Looking at Gauss, who was swearing and his face was filled with blood, he had made a lot of money.
Xu Yun opened his mouth slightly, but stopped talking.
He actually wanted to tell Gauss one thing:
Judging from the historical update rate of Faraday's pigeons, his so-called update is probably just a piece of cake
When Xu Yun was writing novels in his previous life, he also knew several pancake masters, and this kind of thing was not uncommon.
For example, Pei Tu Gou La, Bai Te Man La, Tian Ya Yue Zhao Jin, etc.
Of course.
There are experts at painting cakes, and there are also people with integrity.
For example, Xu Yun himself won praise from a large number of readers with a record of 30,000 updates per day in 2033.
However, judging from normal circumstances, the probability that Faraday is the latter is almost non-existent.
in the original history.
Not to mention ordinary updates, he could even delay updating the more than 3,000-word textbook review that the Royal Society asked him to write for two years.
Therefore, there is a high probability that Gauss was deceived by this pigeon.
But before the words were spoken, Xu Yun had a second thought.
If I told Gauss about this, I'm afraid there would be no chance to exchange for Gauss's manuscript.
Therefore, he stopped what he was about to say, and just laughed a little awkwardly, pretending not to know anything, and turned his attention to the manuscript in front of him.
Then I looked at the manuscripts stuffed in the suitcase.
Gulu——
Xu Yun swallowed hard, a hint of obvious excitement flashing in his eyes.
Oh my God, this is Gauss's manuscript!
Throughout the history of human science.
In the Middle Ages, at home and abroad, everyone who was famous in the industry would basically leave some works written by themselves.
For example, there are Yang Hui's "Yang Hui's Algorithm" in China, Lao Su's "Bencao Illustration" and "New Yixiang Fa Yao" and so on.
Overseas, there are "The Calculation of Sand", "Spiral" and so on.
And with the development of scientific level.
When the timeline moved to the 16th century, manuscripts gradually became an alternative carrier for recording the achievements of scientists.
Compared to ‘works’.
The randomness of the manuscript is undoubtedly much higher, the accuracy and authority less so.
For example, what is recorded above may be the inspiration that a certain scholar came up with, his wild problem-solving ideas, or even random graffiti left when he was bored.
Just like the class notes taken by some students in later generations.
Sometimes in the past month or two, even the creator himself may not be able to understand the contents of the manuscript.
But on the other hand.
But the manuscript may also contain some surprising results.
For example, some creators have solved mathematical answers but are not sure whether there are errors or omissions.
Another example is results that cannot be released due to time constraints, etc.
in human history.
The mathematician with the most surviving manuscripts is Euler, who is also a god who can be called a fool.
He entered the University of Basel at the age of 13 and graduated from university at the age of 15.
Obtained a master's degree at the age of 16, began publishing papers at the age of 19, and became a professor at the Petersburg Academy of Sciences at the age of 26.
He wrote 886 books and papers throughout his life, averaging more than 800 pages per year.
The Petersburg Academy of Sciences was busy for 47 years in sorting out his works.
What's even more embarrassing is.
Euler was almost blind in his right eye when he was 30 years old, and could only see with his left eye.
Then he suffered from cataracts in his left eye. At the age of 59, he underwent surgery to treat cataracts, and his left eye was blinded by the attending doctor. He was completely blind in both eyes.
The result is blindness in both eyes.
Euler still completed several books and more than 400 papers in oral form, solving complex analysis problems such as Yueli that gave Mavericks headaches.
In 1911, the Swiss Natural Science Foundation organized the compilation of "The Complete Works of Euler", which was planned to be published in 84 volumes, each volume being in 4 format - that is, the size of a newspaper, and one volume is close to 300 pages.
As of 2022, this book has been published in 74 volumes and is available on Amazon as "Opera Omnia". (eulerarchive.maa.org/This is the search URL for Euler’s papers, anti-collision appendix)
What's even more embarrassing is.
Can you believe that the extant Euler manuscripts in later generations are not all of Euler's legacy?
Right, not all.
A considerable part of his manuscripts were destroyed in the Great Fire of Petersburg in 1771, and only part of them remains.
So sometimes you really have to wonder if someone is a time traveler, because their resume is so outrageous
On the other hand.
If Euler is a well-deserved writing machine.
Then the title of creator of the most valuable manuscript undoubtedly belongs to Gauss.
Compared with Euler's countless manuscripts, there are actually not many Gauss manuscripts preserved in later generations. There are only 20 notebooks and about 60 letters to and from.
But even if it is only such a small amount of manuscripts, until 2022, when Xu Yun travels through time, there are still a lot of them that have not yet been deciphered.
For example, Manuel Bhargava mentioned earlier.
The project that won him the Fields Medal in 2014 was inspired by the chapters related to quadratic forms in Gauss's "Arithmetic Explorations".
Of course.
The reason why there are many manuscripts in later generations that cannot be summarized is largely due to the sloppy writing of some creators. (sites.pitt.edu/~jdnorton/Goodies/Zurich_Notebook/, this is Einstein’s theory of relativity Manuscript, the words of Lao Ai.)
By the way.
Photocopies of some of these manuscripts can be purchased in bookstores. The more common ones in China are the handwritings of Mr. Qian Lao and Mr. Huang Weilu. Qian Lao’s handwriting is super beautiful.
Same as Euler at the same time.
Some of Gauss's manuscripts were lost after his death, but most of them were due to man-made disasters - Gauss and Weber had a close friendship, and Weber and Gauss's son-in-law were both among the Seven Gentlemen of Göttingen.
Therefore, after Gauss's death, his former residence suffered many illegal break-ins and many things were lost.
In his letter to Dedekind, Riemann mentioned the violent destruction of Gauss's study.
Some of the leaked manuscripts have entered the hands of collectors. In 2017, a Spanish collector returned two notebooks to the University of Göttingen.
This kind of inability to live in peace after death is actually very common in the scientific community. The most unlucky person is actually not Gauss, but Lao Ai:
This great man in the history of science, who competed with a calf for the first place until a dog's brain was about to be beaten out, had his real brain stolen by a doctor named Harvey seven hours after his death and cut into 240 pieces.
It was not until forty-two years after his death that Harvey handed over slices of his brain to Princeton University Hospital.
This is also the real reason why some novels in later generations would ridicule slicing, although it is estimated that many authors who wrote the word "slicing" did not know this.
Think of this.
Xu Yun couldn't help but sigh and returned his thoughts to reality.
He first took out the laboratory gloves from his body - the gloves these days are latex gloves with basic lead carbonate added, and the cost is relatively high, so when doing non-toxic experiments, they basically bring their own and use them repeatedly. .
After putting on gloves.
Xu Yun then bent down and began to look through Gauss's manuscript.
"Random Thoughts on Advanced Analysis."
"Eulerian characteristic number problem in topology"
"Explanation of doubts about the path of complex variable function theory."
Gauss placed many manuscripts in his suitcase, and the names were extremely complex, but Xu Yun's goal was also quite clear:
He only wanted original manuscripts that had been lost or had special significance.
As for non-Euclidean geometry, a manuscript that was not published in 1850 but has been fully formed into a system in later generations, it was definitely not the goal of his trip.
After a while.
Xu Yun's eyes suddenly lit up and he took out a relatively inner manuscript:
"Huh?"
I saw a line written on the seal of this manuscript:
"Calculation of Affinity Numbers".
Affinity number.
The English name of this word is friendlynumber, so it is sometimes translated as friendly number or blind date number.
Its meaning is very simple:
Two positive integers whose sum of all divisors (except themselves) is equal to the other, such as 220 and 284.
Give an example.
Friends who have been to elementary school should all know this.
The divisor of 220 is:
1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, the sum is 284;
The divisor of 284 is:
1, 2, 4, 71, 142, and the sum is exactly 220.
Therefore, 220 and 284 are a pair of affinity numbers.
The word first appeared in 320 BC and originated from ancient Greece, one of the birthplaces of Western civilization.
Pythagoras, the academic giant at that time, had unfathomable research on number theory. He was the proposer of "everything is number".
His disciples were influenced by him and became "obsessed" with the study of numbers, trying to find numbers in everything in the world.
The result is one day.
His disciples had a sudden idea and asked Pythagoras a question:
Teacher, when I make friends, will there be a relationship between numbers?
As a result, Pythagoras said a very famous saying:
Friends are the shadow of your soul. They should be as close as 220 and 284. You are in me and I am in you.
This sentence is the source of all evils in affinity numbers.
After the advent of Affinity Numbers, Master Bi did not rest. Instead, he led the Bi School to take the opportunity to vigorously promote "all things are numbers".
But it's embarrassing.
Leader Bi has been promoting and researching for decades, but the affinity numbers are still only 220 and 284.
Until the death of Master Bi, people's understanding of affinity numbers still stayed at 220 and 284.
What’s even more embarrassing is that in the following hundreds of years, the mathematical community still has not found the second pair of affinity numbers.
So everyone began to suspect that 220 and 284 were two numbers that Master Bi happened to say casually.
As the research interest in affinity numbers wanes, it gradually fades out of people's sight.
Until 850 AD, the Arabian Almighty King mathematician Tabet bin Korah proposed an idea:
There must be more than one pair of affinity numbers among the infinite natural numbers!
Unlike previous mathematicians, he did not intend to sift through the endless natural numbers.
Instead, we start from general rules and try to find a general formula for affinity numbers.
This almighty king gave up the study of all other subjects in order to study affinity numbers, and became bald when he was only in his 20s.
But hard work paid off, and he finally concluded a rule:
a=3X2^(x-1)-1
b=3X2^x-1
c=9X2^(2x-1)-1.
x here is a natural number greater than 1. If abc are all prime numbers, then 2xab and 2xc are a bunch of friendly numbers.
For example, if x=2, then a5, b=11, c=71.
Therefore, 2×2×5×11=220 and 2×2×71=284 are a pair of affinity numbers.
As soon as the conclusion came out, it was proved that Master Bi was not just talking nonsense. Affinity numbers do exist and can be calculated.
From here on, the story starts to get interesting...
since then.
Mathematicians no longer search for affinity numbers without a clue.
Instead, while looking for a simpler formula, we use a large number of calculations to find the affinity number.
But sadly.
In the following 800 years, mathematicians not only failed to optimize the formula of the Almighty King, but also failed to find a new pair of affinity numbers.
That is to say.
2500 years after Pythagoras, no one could find the shadow of the second pair of affinity numbers!
This situation lasted until 1636, when King Fermat stepped onto the stage of history, breaking the historical embarrassment of more than 2,500 years.
This "amateur mathematician" can't stand it anymore. He supports his family during the day and calculates affinity numbers at night. His head is buzzing with calculations.
Finally, when his head was full of gray hairs, he finally found the second pair of affinity numbers:
17296 and 18416.
After Fermat, Descartes also calculated the third pair of affinity numbers:
9437056 and 9363584.
Then comes the appearance of the big-ass, humanoid self-propelled manuscript printer Euler:
In 1747, when he was 39 years old, he found 30 pairs of numbers in one breath!
Then before everyone could react or even have time to applaud, he announced that he had found 30 pairs again.
But at this point, the affinity number froze:
It was not until 1923 that the mathematicians Medaqi and Ye Weiler took the initiative and built a plank road openly and secretly.
They expanded the affinity number to 1095 pairs in one go, the largest of which even reached 25 digits.
Between 1747 and 1923, mathematicians calculated 217 pairs of affinity numbers using only Euler's formula.
Of course.
With the invention of computers, the calculation of affinity numbers has become much simpler.
Just like pi has been calculated to 62.8 trillion digits, the affinity number for later generations has been locked to more than 380,000 digits.
You see, many people have girlfriends, but some people are still single.
Oh, so does Xu Yun, then it’s okay.
all in all.
Under the premise that later generations have calculated a large number of affinity numbers.
What Xu Yun is looking forward to is not how much help Gauss's manuscript will bring to the future, but
Gauss, as a famous prince of mathematics, has he ever done any calculations on affinity numbers?
At least in Xu Yun's perception.
This manuscript is definitely not among the "relics" of Gauss in later generations - at least there is no relevant manuscript in the handwriting that has been made public.
Think of this.
Xu Yun couldn't help but glance at Gauss and said:
"Professor Gauss, do I have to select the manuscript before viewing the content?"
Gauss nodded:
"Of course, subsequent content needs to be paid to watch."
Gauss's answer was as expected by Xu Yun, so he didn't think about bargaining or anything, and immediately replied:
"Then Professor Gauss, this is the first manuscript I chose."
Gauss waved his hand when he saw it, meaning it's up to you.
After getting Gauss's permission.
Xu Yun solemnly took the manuscript to the desk and carefully unsealed it.
The prop to bind the manuscript was a red silk thread. Xu Yun took one end of the thread and pulled it like a shoelace.
Phew——
The manuscript unfolded instantly.
This manuscript is surprisingly thin, only about one or two pages.
Xu Yun picked it up with gloves still on and looked at it carefully.
There are several numbers recorded at the beginning of the manuscript, which are:
220/284、2924/2620、17296/18416、9437056/9363584
There is nothing special about these numbers. They are all affinity numbers calculated by previous generations.
Then there is the formula induced by Euler.
But when Xu Yun continued to glance down a few more times, his breathing suddenly stopped for a few seconds.
I saw a few numbers written in the lower half of the manuscript:
5564/5020
6368/6232
10856/10744
14595/12285
18416/17296
1000452085744/1023608366096
1001583011750/1019368284250
A clear black dot can be seen at the end of the last set of numbers, apparently the mark left by the pen tip.
Under this set of numbers, you can also see a formula:
σ(z)=σ(xy)=1+[σ(x)-1]+[σ(y)-1]+[σ(x)-1][σ(y)-1]=1+σ (x)+σ(y)-2+σ(x)σ(y)-σ(x)-σ(y)+1=σ(x)σ(y)
D(x)=x(1+12+13++1x2)≈x[ln(x/2+1)+r]≈x(lnx-0.116).
In addition, on the right side of the formula, there are several flying letters.
Translated into Chinese characters:
[It’s too simple to count, it’s boring to death].
"."
Xu Yun was speechless for a long time, then raised his head and looked at Gauss.
Gauss blinked:
"What are you looking at?"
Xu Yun gently raised the manuscript in his hand towards him and said to Gauss:
"Professor Gauss, that sentence at the end of your manuscript."
"Oh, you said that."
Gauss recalled for a few seconds, and soon remembered what Xu Yun said, and explained:
"Literally, it took me two days after receiving the Euler manuscript from Joseph. It should have been two days, or three days - anyway, I quickly calculated hundreds of sets of affinity numbers. ”
"Later, I originally wanted to summarize a corresponding formula, but after half of the calculation, it felt too simple, so I put it aside."
"Oh, by the way, Bornhard also figured out this formula three years ago. His evaluation is that it is enough to have a good hand."
Xu Yun:
"."
The Joseph mentioned by Gauss was Joseph Louis Lagrange, who was also a disciple of Euler and a mathematician who left his name in history.
His relationship with Euler is almost that of Riemann and Gauss.
Euler - Lagrange - Cauchy, and Gauss - Dirichlet - Riemann, these are two famous inheritance schools in modern mathematics.
Also in history.
Lagrange was also one of the inheritors of Euler's manuscript, so it was normal for him to send a letter to Gauss.
Just
Gauss's words are too damn offensive, right?
To know.
Even in 2022, when Xu Yun traveled back in time, there is still no unified affinity number formula in the mathematical world.
Whether it is Euler or Yeweiler, their formulas have a certain error rate - for example, Euler missed the calculation of the group of 1184/1210, which was not calculated until 1867 by an Italian child prodigy.
The name of this child prodigy is Paganini. Every time he thinks of this name, Xu Yun always thinks of Panini
The selection of affinity numbers in later generations mainly depends on divisors and comparisons, that is, if the conditions are met, YES will be output, and otherwise, NO will be output.
To put it harshly.
The essence of screening for future generations is actually the exhaustive method.
As a result, in the era of 1850, both Gauss and Riemann derived the standard formula of affinity numbers?
However, considering the achievements of these two people in history, and Euler has deduced part of the affinity number formula.
Well, it doesn't seem surprising that they were able to do this.
at the same time.
This can be regarded as solving a mystery in the history of mathematics:
Before the invention of computers, almost every school of mathematics devoted a lot of energy and time to affinity numbers.
But only Gauss's Göttingen school of mathematics was excluded.
Neither Gauss himself nor Riemann, Jacobi, Dedekind or Dirichlet left any works or records on the study of affinity numbers.
This is actually a very strange phenomenon, as inconsistent as if later generations of quantum theory masters did not study perturbation theory.
Now with these words of Gauss, everything is finally revealed:
They co-authored that they had already cracked the mystery of affinity numbers, but they thought it was too simple and didn’t bother with it.
Then Gauss glanced at Xu Yun, who was still unfinished.
After pondering for a moment, he took the initiative to go to the suitcase and rummage around a few times.
soon.
He took out another slightly thicker manuscript, handed it to Xu Yun, and said:
"Luo Feng, since you are interested in affinity numbers, this manuscript may suit your taste." (End of Chapter)