Great Country Academician

"Using the regularity of the boundary points of the Dirichlet function to construct a function field with a regular boundary, and then introducing the curve equation by expanding the field to limit the concept of the dual reduction group"

In the auditorium of Wenjin International Hotel, Artur Avila muttered a few words to himself, his eyes suddenly brightened, and he looked at Xu Chuan excitedly.

"Xu, you really deserve to be hailed as the strongest genius in the history of mathematics. You are so powerful. Using this method, you may really be able to constrain and determine the functor properties of some automorphic groups."

Xu Chuan looked embarrassed, what the hell is this "the strongest genius in the history of mathematics"? Who gave him this name?

However, during the exchange and discussion, he didn't pay much attention to this. He nodded, and he continued following the words of Professor Artur Avila:

"More than that, the first instance of the Langlands functor conjecture to be verified is the functor between the automorphic representation of GL2 on the algebraic number field and the representation of the multiplicative subgroup of the quaternion algebra."

"The functoriality proved in this classic work also proposes the relationship between the original form of Artin's conjecture and the functoriality conjecture, and Artin's conjecture is also reformulated as the two-dimensional complex representation of the Galois group and the GL2 automorphic group represents a functorial conjecture between

"Therefore, Artin's conjecture points out that the Artin L functions constructed on Galvaro groups are holomorphic, and Langlands conjectures that these Artin L functions should be L functions represented by automorphic groups in essence."

Hearing this, Professor Artur Avila fell into deep thought, but after a while, he suddenly came to his senses, and said with half doubt and half certainty:

"If Artin's conjecture can be proved, then can Artin's L function take a big step forward on the Langlands conjecture?"

Xu Chuan nodded and said, "From the current theory, this is indeed true."

Immediately, he shook his head again and said, "But."

"But it's too difficult to solve Arting's conjecture." Professor Artur Avila sighed and added what Xu Chuan hadn't finished.

Xu Chuan acquiesced and did not speak again.

Artin's conjecture, also known as New Mason's conjecture, is a derivative of the famous Mason's conjecture, and it is a conjecture about prime numbers.

If you haven't heard of Artin's conjecture and Mason's conjecture, most people should have heard of the familiar Goldbach's conjecture.

They are all a type of conjecture, which can be said to be derived from prime numbers.

In mathematics, the earliest people come into contact with are natural numbers like 0, 1, 2, 3, and 4.

Among such natural numbers, if a number is greater than 1 and cannot be divided by other natural numbers (except 0), then this number is called a prime number, also called a prime number.

Numbers greater than 1 but not prime are called composite numbers. 1 and 0 are special, neither prime nor composite.

As early as 2,500 years ago, people at that time noticed this peculiar phenomenon, and Euclid, the father of geometry, an ancient Greek mathematician, proposed a very classic prove.

That is: Euclid proved that there are infinitely many prime numbers, and proposed that a small number of prime numbers can be written in the form of "2^p-1", where the exponent p is also a prime number.

This proof is called "Euclidean prime number theorem", which is one of the most basic classical propositions in number theory.

The classics never go out of date. Subsequent mathematicians derived various conjectures about prime numbers when studying the "Euclidean Prime Number Theorem".

Starting from the Mersenne prime conjecture, Zhou's conjecture, twin prime conjecture, Ulam spiral, Gilbreth conjecture, and finally the extremely famous Goldbach conjecture and so on.

There are many conjectures derived from prime numbers, but most of them have not been proved.

The new Mersenne prime number conjecture that Xu Chuan talked with Professor Artur Avila is a conjecture derived from prime numbers, also called Artin's conjecture, which is an upgraded version of the original Mersenne prime number conjecture.

Among the many conjectures of prime numbers, the difficulty is comparable to that of the twin prime conjecture, second only to the famous "Goldbach's conjecture".

[New Mersenne prime number conjecture: For any odd natural number p, if two of the following statements are true, the remaining one will be true:

1. p=(2^k)±1 or p=(4^k)±3

2. (2^p)- 1 is a prime number (Mersen prime number)

3. [(2^p)+ 1]/ 3 is a prime number (Wagstaff prime number)]

The new Mersenne prime number conjecture has three problems, and the three problems are closely related. If two of them can be proved, then the remaining one will be established naturally.

In the history of scientific development, the search for Mersenne prime numbers was used as an important indicator to detect the development of human intelligence in the era of hand calculation records.

Just like today's IQ test questions, the more Mersenne primes that can be calculated, the smarter the person is.

Because Mersenne primes seem simple, but when the exponent P value is large, its exploration not only requires advanced theory and skilled skills, but also requires arduous calculations.

The most famous, Euler, known as the "God of Mathematics", proved that 2^31-1 is the eighth Mersenne prime number by mental arithmetic in the case of blindness;

This prime number with 10 digits (ie 2147483647) was the largest known prime number in the world at that time.

It is good for ordinary people to be able to add, subtract, multiply and divide three-digit numbers, but Euler can push numbers to the billion level with mental arithmetic. This terrifying calculation ability, brain response ability and problem-solving skills can be said to be worthy of the "chosen child" reputation.

In addition, in 2013, a research team led by mathematician Curtis Cooper of the University of Central Missouri in the United States discovered the largest Mersenne prime so far by participating in a project called "Internet Mersenne Prime Search" (GIMPS) ——2^57885161-1 (2 to the 57885161 power minus 1).

The prime number is also the largest known prime number, with 17,425,170 digits, which is 4,457,081 digits more than the previously discovered Mersenne prime number.

If it were printed in ordinary 18-point standard font, it would be more than sixty-five kilometers long.

Although this number is very large, it is very small in terms of mathematics.

Because the "number" is infinite, the number has the concept of infinity. In mathematics, no one knows how many prime numbers there are after the number 2^57885161-1 (2 to the 57885161 power minus 1).

This has lasted for thousands of years, and it is the largest exploration journey in the history of mathematics: how many Mersenne primes are there, and whether they are infinite or not? Up to now, no one has yet given the answer.

Proving the new Mersenne prime conjecture is no less difficult than the Weyl-Berry conjecture that Xu Chuan proved before.

So far, the most difficult proof of the prime number conjecture in the mathematical community is only the weak Goldbach conjecture.

That is: [Any odd number greater than 7 can be expressed as the sum of three odd prime numbers. 】

In May 2013, Harold Hoofgot, a researcher at the Ecole Normale Supérieure de Paris, published two papers announcing the complete proof of the weak Goldbach conjecture.

In addition, in the same year, Professor Zhang Yitang, a mathematician in Huaguo, also made considerable progress in the proof of the prime number conjecture.

His paper "Bounded Distances Between Prime Numbers" was published in the "Annual Journal of Mathematics", which solved the problem that has plagued the mathematics community for a century and a half, and proved the weakening situation of the twin prime conjecture.

That is: find that there are infinitely many pairs of prime numbers whose difference is less than 70 million.

This is the first time someone has proved that there are infinitely many pairs of prime numbers whose distance is less than a certain value.

But for the mathematics community, whether it is the weak Goldbach conjecture or the weak twin prime number theorem, they are just the prelude to the peak.

They are like a resounding national anthem before a climber climbs Mount Everest, which can give climbers courage to a certain extent, but it is unrealistic to expect to climb Mount Everest and stand on the summit.

"Xu, will you try to develop in the direction of number theory?"

After a slight silence in the atmosphere, Professor Artur Avila looked up at Xu Chuan.

If this youngest genius in the history of mathematics develops in the direction of number theory, maybe he has the opportunity to pick a huge fruit in the field of prime numbers?

He can't say for sure, after all, who can be sure about this kind of thing.

Artur Avila really wanted to see the day when Goldbach's conjecture was confirmed, but he didn't want this new star in the mathematics world to plunge into it for several years or even decades without making achievements.

Prime numbers have been developed for thousands of years, and countless mathematicians have rushed into this huge pit one after another, although they have proved many conjectures and solved many problems.

But from beginning to end, the most difficult problems have not been solved.

Even, there is no hope of a solution.

However, if Xu Chuan continues to study spectral theory, functional analysis, and Dirichlet functions, he dare not say that he will be able to make greater contributions than the Weyl-Berry conjecture, but he will definitely be able to further expand the boundaries in these fields and expand math range.

But if it is transferred to number theory, it is not sure.

Not every genius belongs to Tao Zhexuan. At present, Xu Chuan's mathematical talent is indeed higher than Tao Zhexuan's, but no one knows what will happen after crossing fields.

Xu Chuan did not give Avila a definite answer. In the past year, he had indeed read a lot of books related to number theory, but number theory was not in his follow-up study and research arrangement.

He prefers functions and analysis that can be practically applied to solve physical problems, while number theory mainly studies the properties of integers, which can be regarded as pure mathematics.

Of course, with the development of mathematics today, it is impossible to say that any field of mathematics is pure mathematics, and it can always be linked with other fields.

For example, in statistical mechanics, the partition function is the basic mathematical object of research; while in the analytical theory of prime number distribution, the zeta function is the basic object.

Thus, this unorthodox interpretation of the zeta function as a partition function points to a potentially fundamentally meaningful connection between the distribution of prime numbers and this branch of physics.

It's just that at present, the application of number theory to the field of physics is still relatively vacant, far less extensive than mathematical analysis, function transformation, and mathematical models.

So Xu Chuan is not very inclined to invest a lot of energy and time in the field of pure number theory.

But it is certain to study and study number theory.

Because number theory is not only pure number theory, but also various branches such as analytic number theory, algebraic number theory, geometric number theory, computational number theory, arithmetic algebraic geometry and so on.

These branches are all extended from pure number theory, that is, elementary number theory combined with other mathematics.

For example, analytic number theory is to use calculus and complex analysis (that is, complex variable functions) to study number theory about integer problems.

The things he talked with Professor Avila tonight have something to do with analytic number theory.

Because the method of analytic number theory includes not only the circle method, the sieve method, etc., but also the modular form theory related to elliptic curves, etc. Since then, it has developed into the theory of automorphic forms, which is connected with representation theory.

Therefore, having a certain number theory foundation is of great help to other mathematics learning.