Into Unscientific

Chapter 308 Gauss' Treasure (Part 2) (8.4K)

"."

In the study.

Looking at the brand new manuscript that Gauss handed in front of him, Xu Yun couldn't help showing curiosity.

What will be the content of this header?

To know.

In the field of mathematics, affinity numbers belong to a branch of number theory.

If you really want to count the "relatives" that can be matched with it, there are too many examples that meet the conditions.

For example, prime numbers, equal sum numbers, isolated numbers, common sum numbers, etc. are a lot of

Even if you insist on talking.

Non-Euclidean geometry can 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, relying on Gauss' introduction alone, Xu Yun really couldn't guess the content of this manuscript, and he could only know it by reading it himself.

Then he stretched out his hands and took the manuscript carefully.

Then he thought of something again, stopped and asked Gauss:

"Professor Gauss, you gave me this manuscript, just read it."

As a result, before Xu Yun finished speaking, Gauss ruthlessly dismissed his thoughts:

"Of course it must be included in one of the five volumes."

Xu Yun could only shrug.

Well, the card logic bug fails.

But overall the problem is not big, after all, the chance of these five volumes of manuscripts is a surprise in itself.

Then he looked at the outside of the manuscript, and found that the manuscript was only tied with a red ribbon, and he didn't see the seal with the general content similar to the affinity number.

See this situation.

Xu Yun's eyes were fixed immediately, and the importance in his heart increased a bit:

The manuscript can be found without the title index, which shows that it must have an unusual position in Gauss's heart, at least there is no need to rely on the seal for memory reminders.

Think here.

Xu Yun couldn't help untying the ribbon a little faster, as if he was untying a shoelace.

Well, untie your shoelaces and don't think too much about it.

After half a minute.

A roll of flattened manuscript paper appeared in front of Xu Yun.

Xu Yun pinched the two corners of the upper half of the manuscript paper, picked it up as if urging the party to carry the author upside down, and read it line by line.

After a few seconds.

Xu Yun's pupils shrank suddenly, and the manuscript in his hand almost fell to the ground in shock!

At the beginning of this manuscript paper, there was a line written:

"Proof of the Non-Existence of Odd Perfect Numbers"

The correct reading of this title is [Proof about/Odd Perfect Number/Non-existence/], the most critical core of which is the two words in the middle:

Odd perfect number does not exist.

Students who understand number theory should know it.

If these two words appear at the same time in 2022, it is destined to cause a major earthquake in the mathematics world.

mentioned earlier.

In 2022, when Xu Yun traveled, the status of affinity numbers in the mathematics world has always been a little awkward:

on the one hand.

Affinity numbers can be exhaustively listed by computer, and the sum of divisors can be compared like a production line.

If the condition is met, the output is YES, otherwise it is NO, and it can be done with one key.

As of 3:34 am on August 15, 2022, the number of affinities discovered has exceeded 11,994,387.

The longest pair of numbers is more than 24 million digits—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 well, you can think of "bit" as a word.

24 million digits, which is equivalent to 24 million words of online novels.

If the author lists this number, the number of words in our book can immediately jump to a few meters before the starting point.

In fact, this is not the most outrageous, the pi mentioned in the last chapter is the scariest - it has been calculated to 100 trillion digits. (Thanks to the readers for their corrections. I checked that the 62 trillion records have indeed been refreshed, and it has only been less than eight months, which is too fast)

The record was set by Google Cloud engineer Emma Haruka Iwao, a neon person.

Ta used 25 Google virtual machines, spent 158 ​​days before and after, and finally set this record in June this year.

This is also the project leader who calculated 31.4 trillion pi in 19 years, but compared with his achievements, this person's orientation is also quite subtle:

It is not difficult to see from the previous ta that this boss is a supporter of lesbians who are physically female and psychologically male.

So Xu Yun sometimes wonders, do capable people like to add buffs to themselves these days?

ok, the topic will return to the original place.

Since the computer can screen out so many affinity numbers, why is it still called embarrassing?

the reason is simple.

That is, the specific law of the affinity number has not been completely deciphered, and the computer only relies on the exhaustive method.

This method has led to the emergence of another part of 'variant' and unknown numbers among these affinity numbers.

Say 12496.

You add up its divisors and you get the number 14288.

Then add up the divisors of 14288 to get 15472;

Then continue the process.

15472 will become 14536

14536 will become 14264

14264 would become

12496.

That's right.

After five changes, it was exactly back to the starting point.

This number is called a communicative number.

Because its circle of friends is wider than the number of affinity, or the number of blind dates, some people call it Neptune Number.

In addition to the social number, there is another number that is also extremely special.

That is a perfect number, also known as a perfect number.

The concept of this number is actually very simple:

When you add their divisors, you get them themselves.

The smallest example is 6.

The divisors of 6 are 1, 2 and 3, and 1+2+3=6.

After that is 28, because 28=1+2+4+7+14.

The next perfect number of 28 is 496, and then there is a relatively large leap to 8128.

As for further

It's getting more and more absurd.

For example, the next perfect number of 8128 is 33550336, followed by 8589869056, followed by 137438691328.

The one behind is 2305843008139952128, which looks like an ID card.

By the time Xu Yun crossed, there were only 51 complete numbers.

The largest known perfect number was discovered in 2018, with 49,724,095 digits and as many as 111,577,0321 approximations.

It is equivalent to a novel of 49 million words, fully twice the maximum affinity number above. Adding the two together, only "Universal Giant School Flash" has more than two words in the entire network

This is actually a very nerve-wracking thing:

Think about it.

Its 1115770321 divisors add up to exactly equal to itself.

Therefore, the reason why many people in later generations think that the mysteries of the universe are hidden in mathematics is not because they are paying attention to their own industry, but because some numbers are really exquisite to the extreme.

In addition, the subject of mathematics also reflects the dark and cruel reality of the universe from a philosophical point of view—if you don’t know it, you won’t, at most you can get one point for writing a solution, and even gods can’t save you.

cough cough

In addition to the characteristics of divisors, perfect numbers have two special features:

One is that all the perfect numbers discovered so far have a one-to-one correspondence with Mersenne primes without exception.

That is, there are as many perfect numbers as there are Mersenne primes found.

Now the relevant calculation is performed by a project team called GIMPS. In 14 years, a total of 10 Mersenne prime numbers or perfect numbers have been found.

The Huaxia National Team currently ranks eighth in the contribution of this project group, with a total contribution of about 1.5%.

By the way, share a URL called equn.com, which is the official website of Huaxia Distributed Computing Headquarters.

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 appetite and apply to join.

Except that the perfect numbers are in one-to-one correspondence with the Mersenne primes.

The second peculiarity of 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 of odd perfect numbers in 2022 is the proof proposed by Austin Ohr:

If there is an odd perfect number, its form must be 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.

And then it's gone.

That's right, it's gone - there is basically no progress in the theoretical direction of odd perfect numbers in the mathematical community.

Of course.

This means that no results were born, and it does not mean that everyone has given up related computing work.

It's just that Xu Yun didn't expect that.

Gauss seems to be a problem that has caused headaches and even baldness for countless people in later generations.

Solved in 1850?

Mom!

Xu Yun dared to bet that he had a manuscript that did not exist at all. There must not be such a manuscript among the "relics" of Gauss in the future!

Think here.

Xu Yun couldn't restrain the excitement in his heart, and began to look it up seriously.

The first volume of the manuscript is not a calculation and derivation process, but a diary-like essay.

"1831 Alley, Clear September, Chapter 7 of Faraday's update, Dynamos continue to push the next line of human development."

"On September 15th, after finishing Mina's funeral, I felt extremely sad."

"After seven days of silence, Therese's recitation suddenly came from outside the window, [Mr. Fat Fish helped the 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, which gave me the courage to look forward again."

"It just so happened that Dirichlet was visiting, and I happened to see the "Unsolved Mysteries of Mathematics" revised by the University of Würzburg in his hand, and I became more and more playful."

"So I wrote down a few small pieces of paper, folded them into balls, and asked Trezer to randomly select one of them. The topic on it was 'Does odd-perfect numbers exist?'"

"Finally, it took four hours and thirty-five minutes to write this manuscript, put on my pants, and evaluate 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 huge and obvious word appeared in front of him:

untie.

untie:

"It's well known."

"A positive integer n is an even perfect number if and only if n=2m1(2m1)n=2^{m-1}(2^{m}-1)n=2m1(2m1) where m, 2 m1m, 2 ^{m}-1m, 2^m1 are all prime numbers."

"If p is a prime number and a is a positive integer, then:"

"σ(pa)=1+p+p++p^a={p^(a+1)1}/p-1."

"Let a positive integer n have prime factorization n=p^(a1/1)p^(a2/2)p^(a3/3).p^(as/s)."

"Since the factors and functions σ are multiplicative, then:"

"σ(n)={p^(a1+1/1)-1}/{p1-1}·{p^(a2+2/1)-1}/{p2-1}·{p^( a3+3/1)-1}/{p3-1}·{p^(as+s/1)-1}/{ps-1}=s∏j1·{p^(aj+j/1) -1}/{pj-1}. (S should be above ∏ and j=1 below, but the starting point does not support it.)”

"And because p is an odd prime number, a is a positive integer, and s≥1."

"So there is {p^(a1+1/1)-1}/{p1-1}\u003c{p^(a1+1/1)}/{p1-1}=(p1)/(p1-1) p^(a1-1/1)≠2p^(a1-1/1)≠2p^(a1-1/1)."

"{p^(a2+2/1)-1}/{p2-1}\u003c{p^(a2+1/1)}/{p2-1}=(p2)/(p2-1)·p ^(a2-2/1)≠2p^(a2-2/1)≠2p^(a2-2/1)"

"{p^(as+s/1)-1}/{ps-1}\u003c{p^(as+1/1)}/{ps-1}=(ps)/(ps-1)·p ^(as-s/1)≠2p^(as-s/1)≠2p^(as-s/1)"

"In square numbers, the sum of their continuous additions, multiplied by 6, and some are 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 2 powers and 5 powers, there is a form of 2 times 3 times 5 times 7... In mathematical calculations, on the contrary, it is to calculate the sum of continuous additions, and 1 power, 2 times The power is the same, write out its calculation form, that is, add 1 and subtract 1 to an even number, and it can be written as a prime number and a composite number.”

"So σ(n)≠2{p^(a1+1/1)-1}/{p1-1}·{p^(a2+2/1)-1}/{p2-1}·{p ^(a3+3/1)-1}/{p3-1} {p^(as+s/1)-1}/{ps-1}."

"That is, σ(n)≠2n, where n is an odd number greater than 1, and σ(1)=1, σ(1)=1."

"so."

"There is no odd perfect number." (In fact, the last step is impossible to pass. It was a trick, so don't go into it. The inspiration is from 10.3969/j.issn.1009-4822.2009.02.003)

Look at the last sentence where the pen is written.

Xu Yun was silent for a long time.

The thousand words in my heart finally turned into a long sigh.

This is Gauss

A man who stood at the pinnacle of the history of mathematics throughout the ages, a German who conquered a territory wider than a certain mustache.

Xu Yun was mesmerized by a volume of manuscripts that looked like essays

suddenly.

Xu Yun thought of what Gauss said to him before:

"I don't perform miracles, because I am a miracle."

This little old man, with his talent and intelligence, abruptly became one of the highest peaks in the history of mathematics.

Even in Xu Yun's later life, there is still no one who can hold his own.

anyway.

Mavericks, Lao Su, Lao Jia, Faraday, plus today's Gauss.

Xu Yun can't remember clearly, this is the first time he has lamented the wisdom of the sages.

If given the chance, I would really like to write a novel about my experience.

And just when Xu Yun's mood was soaring.

Gauss' voice suddenly sounded in his ear:

"Student Luo Feng, how is the quality of this manuscript?"

Only then did Xu Yun pull 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 piezoelectric ceramic preparation techniques."

"Perhaps hundreds of years later, technology has developed to an astonishing level. Human beings can fly to the world, but they will still be amazed by your wisdom."

Xu Yun's words did not contain any exaggeration, because he really thought so.

The discovery of the piezoelectric effect is the Curie brothers. To be honest, this technology can only be regarded as quite satisfactory.

There are many technologies that can replace piezoelectric ceramics in later generations, but piezoelectric ceramics have the lowest cost, the most mature technology, and relatively simple preparation difficulty.

The odd complete number of manuscripts is different.

It is a problem that has plagued the mathematics community for nearly 350 years!

Although its status in later generations is not as good as the Riemann conjecture or the Hodge conjecture, it is also a very important research direction.

Although no results have been published, it 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 breakthrough in the country, but it cannot be denied that the country has not invested a lot of energy and financial resources in it.

So in Xu Yun's view.

A volume of manuscripts that can solve odd perfect numbers is indeed worth ten piezoelectric ceramics.

And opposite him.

Seeing that Xu Yun, the "offspring of a fat fish", praised him so much, Gauss immediately had an uncontrollable smile on his face - based on his life experience, it is natural to see whether Xu Yun's exaggeration is true or false.

He waved his hands with a 'modest' face, smiled and said to Xu Yun:

"Student Luo Feng, it's too good to be true. It's just a relatively ordinary achievement, not that high-value—by the way, can you say those words again when Michael is present?"

Xu Yun: ".?"

Afterwards, he solemnly put away the manuscript again, and placed it next to the Affinity Number manuscript.

Then Xu Yun was planning to search for the next volume of manuscripts, but when he was about to start, a flash of light suddenly flashed in his mind.

He likes to eat watermelon very much, but he doesn't know how to pick it, which belongs to the situation of vegetables and love to play.

So every time he goes to the supermarket, he likes to ask those aunts and aunts for help.

With a good voice, most aunts will help with a little effort.

Although there are occasional overturns due to aunt's lack of skill, most of the time the melons picked out are much better than his own hand-picked ones.

And the current selection of manuscripts is the same as picking watermelons

And this is far more than just an aunt visiting the market, he is a melon farmer who grows watermelons!

What manuscripts are helpful, Gauss must be better than Xu Yun!

Think here.

Xu Yun quickly turned his head and looked at Gauss expectantly, with obvious meaning:

Boss, can you help me choose another volume?

Even though Gao Sidang understood Xu Yun's thoughts, he hesitated for a moment, shook his head and said:

"Student Luo Feng, it's an exception for me to give you five volumes of manuscripts as a present. You still want me to choose them yourself. Isn't that a little bit of an overreach?"

"I won't give any more comments in the future. It's up to you what manuscript you can pick."

Looking at the resolute Gauss, Xu Yun thought for a while and said:

"Professor Gauss, isn't Mr. Faraday holding a press conference for his new work in a few days, and school leaders such as Mr. William Whewell will also show up. I can take advantage of the presence of the media to praise your manuscript and Fat Fish Ancestors do not distinguish between uncles"

Xu Yun didn't finish his sentence.

There was a flash in front of his eyes, leaving only an afterimage and Gauss' voice in the air:

"Stand here and don't move around, I'll pick out some manuscripts for you!"

Xu Yun:

"."

Boss, you fucking should be more reserved

After coming to the side of the suitcase.

Gauss leaned down slightly, his eyes kept scanning the suitcase.

Which ones should I choose?

A few seconds passed.

Suddenly his eyes lit up, he took out two volumes of thicker manuscripts, dusted off the non-existent dust, and handed them to Xu Yun:

"Student Luo Feng, if nothing else happens, you should be interested in these two volumes of manuscripts."

Xu Yun still took it with both hands and checked the external situation.

Both volumes of manuscripts have the same affinities as the first volume, with the relevant labels written:

"Research on Superimposed Light Fields"

"Operator Problems in Flow Regime Metrics"

Then Xu Yun spread them out on the desk as usual, and looked at them seriously.

For a later generation like Xu Yun, neither book is too difficult.

For example, "Research on Superimposed Light Fields" records Gaussian research on Fresnel diffraction, with some additional topological charge and azimuth data.

If someone studies in this direction, he will have some accomplishments in the transmission of the optical fiber output.

"Operator Problems in Flow Metrics" is a bit more complicated.

It involves the prototype of non-Euclidean geometry and Riemannian geometry, and is adapted to the common derivative operator of the Cartesian system.

The difficulty of getting started is much higher than that of "Research on Superimposed Light Fields". It can be said that it is the first achievement of Minkowski space and Rayleigh approximation.

Riley is only eight years old now, and Minkowski is 14 years old.

It is really amazing that Gauss was able to study this far ahead of them.

In addition, this manuscript also determines the order of the tensor. After Gauss passes away, this manuscript will definitely bring great inspiration to Riemann's work.

But admiration is admiration.

At this time, the fluctuation in Xu Yun's heart was not as great as when he saw the second volume of manuscripts.

because

Whether it is "Research on Superimposed Light Fields" or "Operator Problems of Manifold Metrics".

The quality of these two manuscripts is obviously beyond doubt, but they have not been lost in later generations, and they are also one of Gauss's few manuscripts that have been thoroughly studied.

In this situation.

It is impossible for Xu Yun to reach the level of 'ecstatic' no matter what.

Of course.

It cannot be said that Gauss has slighted Xu Yun.

On the contrary, the gold content of these two volumes of manuscripts is actually very high.

If they had appeared in 1850, they would probably have caused more repercussions than odd perfect numbers—especially the latter, which is the prototype of fluid geometry.

The reason for Xu Yun's and Gauss's unequal ideas is not the quality of the manuscripts, but the differences in their respective times.

The completeness of the knowledge theory of the era has led to the fact that the two do not look at the problem on the same plane.

However, the regrets in my heart were regrets, and Xu Yun did not show any other complicated expressions.

I still gratefully accept these two volumes of manuscripts.

After all, this is Gauss' intention. For Gauss today, these two volumes of manuscripts can be regarded as a half-baked result.

Four volumes of manuscripts have been selected.

Only the last volume remains undecided.

For the last volume, Xu Yun still asks Gauss to choose.

"Is it the last volume?"

Gauss stood beside the suitcase, his eyes quickly swept through the suitcase.

Which manuscript volume should I choose for Xu Yun?

The core manuscript of Non-Euclidean Geometry has been given to Xiaomai himself, and with the relationship between Xiaomai and Xu Yun, Xu Yun must also be able to see that manuscript.

Therefore, manuscripts related to non-Euclidean geometry can be excluded.

Do you want to choose the periodic calculation of the double-needle function?

Or astronomical observations?

Or just choose the geometric representation of the quadratic model function that I completed last year?

Neither seems appropriate.

A few seconds passed.

Gauss suddenly thought of something.

By the way, that thing!

He bent down and slowly picked up a letter that was placed on a certain mezzanine.

Then Gauss put the letter in his palm, and some old fingers slowly stroked the envelope, with hesitant expressions in his eyes.

Xu Yun noticed.

This look of Gauss is not reluctance, but some

sad?

Xu Yun rubbed his eyes. He wondered if he had read it wrong—how could there be such an expression on Gauss' face?

After two full minutes.

Gauss sighed, handed the letter to Xu Yun with a complex expression, and said:

"Student Luo Feng, if nothing else happens, the first four volumes of manuscript should be enough for you to study for a long time."

"So the last volume of manuscript I selected for you is not some undisclosed intellectual achievement, but this letter"

"letter."