Chapter 143 The strongest genius in the history of mathematics
"Using the regularity of the boundary points of the Dirichlet function to construct a function domain with regularity boundaries, and then introducing curve equations by expanding the domain to restrict the concept of dual reduction groups"
In the auditorium of Wenjin International Hotel, Artur Avila's eyes suddenly brightened after muttering a few words, and he looked at Xu Chuan excitedly.
"Xu, you are indeed known as the strongest genius in the history of mathematics. You are so powerful. Using this method, you might be able to constrain and determine the functor properties of some automorphic groups."
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Xu Chuan looked embarrassed. What the hell is going on with this 'most powerful 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 followed Professor Artur Avila’s words:
"Not only that, the first verified instance of Langlands' functority conjecture is the functority between the automorphic representation of GL2 on the algebraic number field and the representation of the multiplicative subgroup of quaternion algebra."
"The functority proved in this classic work also proposed the relationship between the original form of Artin's conjecture and the functority conjecture. Atine's conjecture has also been reformulated as a two-dimensional complex representation of the Galois group and the GL2 automorphic group Functority conjecture between representations.”
"Therefore, Artin's conjecture points out that the Artin L function constructed on the Gavarro group is holomorphic, and Langlands conjecture that these Artin L functions should actually be L functions represented by the automorphic group."
After hearing this, Professor Artur Avila fell into deep thought, but after a while, he suddenly woke up and said with half doubt and half certainty:
"If Artin's conjecture can be proved, then it will be a big step forward for Artin's L function in terms of Langlands conjecture?"
Xu Chuan nodded and said: "From the current theoretical point of view, this is indeed true."
Then, he shook his head again and said: "But."
"But it is too difficult to solve Artin's conjecture." Professor Artur Avila sighed and completed what Xu Chuan had not finished.
Xu Chuan acquiesced and said nothing more.
Arting's conjecture, also known as the new Mersenne conjecture, is a generalized derivative of the famous Mersenne conjecture, which is a conjecture about prime numbers.
If you have not heard of Artin's conjecture and Mason's conjecture, then most people should have heard of the familiar Goldbach's conjecture.
They are all conjectures of the same type, which can be said to be derived from prime numbers.
In mathematics, the earliest people come into contact with are natural numbers such as 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 that are larger than 1 but are not prime numbers are called composite numbers. 1 and 0 are special and are neither prime numbers nor composite numbers.
As early as 2,500 years ago, people at that time noticed this strange phenomenon. Euclid, the ancient Greek mathematician and the father of geometry, proposed a very classic idea in his most famous work "Elements of Geometry". 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’ and is one of the most basic classical propositions in number theory.
Classics never go out of style. When subsequent mathematicians studied the ‘Euclidean Prime Number Theorem’, they derived various conjectures about prime numbers.
Starting from Mersenne's prime number conjecture, to Zhou's conjecture, twin prime number conjecture, Ulam's spiral, Gilbreth's conjecture to the extremely famous Goldbach's conjecture and so on.
There are many conjectures derived from prime numbers, but most of them have not been proven.
The new Mersenne prime conjecture that Xu Chuan and Professor Artur Avila discussed is a conjecture derived from prime numbers. It is also called Artin's conjecture and is an upgraded version of the original Mersenne prime conjecture.
Among the many prime number conjectures, the difficulty is the same as the twin prime number conjecture, second only to the famous 'Goldbach's conjecture'.
[New Mersenne Prime 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 (Messenne prime number)
3. [(2^p)+1]/3 is a prime number (Wagstaff prime number)]
The New Mersenne Prime Conjecture has three problems, and the three problems are closely related. If two of them can be proved, then the remaining one will naturally be true.
In the history of scientific development, the search for Mersenne prime numbers has been an important indicator of human intelligence development in the era of hand calculations.
Just like today's IQ test questions, the more Mersenne primes that can be calculated, the smarter the person is.
Although the Mersenne prime number seems simple, when the exponent P value is large, its exploration requires not only advanced theory and skillful skills, but also arduous calculations.
The most famous one, Euler, known as the "God of Mathematics", proved that 2^31-1 is the 8th Mersenne prime number by mental arithmetic while blind;
This 10-digit prime number (ie 2147483647) was the largest known prime number in the world at that time.
Ordinary people are very good at adding, subtracting, multiplying and dividing three-digit numbers, but Euler can push the numbers to the billions in mental arithmetic. This terrifying calculation ability, brain reaction ability and problem-solving skills can be said to be worthy of the "Chosen One" reputation.
In addition, in 2013, a research team led by mathematician Curtis Cooper of the University of Central Missouri discovered the largest Mersenne prime so far by participating in a project called the "Grand Internet Mersenne Prime Search" (GIMPS). ——2^57885161-1 (2 raised to the power of 57885161 minus 1).
This prime number is also the largest known prime number so far, with 17425170 digits, which is 4457081 digits more than the previously discovered Mersenne prime number.
If printed out in regular 18-point standard font, it would be over sixty-five kilometers long.
Although this number is very big, in terms of mathematics, it is very small.
Because "number" is infinite, and numbers have the concept of infinity, mathematically speaking, after the number 2^57885161-1 (2 raised to the power of 57885161 minus 1), no one knows how many prime numbers there are. This journey of exploration has lasted for thousands of years and is the largest in the history of mathematics: how many Mersenne primes are there and whether they are infinite? As of now, no one can give the answer.
Proving the New Mersenne Prime Conjecture is no less difficult than the Weyl-Berry Conjecture that Xu Chuan had previously proved.
So far, the most difficult proof of prime number conjecture in mathematics is only the weak Goldbach's 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 in Paris, published two papers, announcing that he had completely proved the weak Goldbach conjecture.
In addition, in the same year, Chinese mathematician Professor Zhang Yitang also made considerable progress in proving the prime number conjecture.
His paper "Bounded Distances Between Prime Numbers" was published in the "Annals of Mathematics", solving a problem that has plagued the mathematical community for a century and a half, and proving the weakening situation of the twin prime conjecture.
That is: It is found that there are infinite pairs of prime numbers whose difference is less than 70 million.
This is the first time anyone has shown that there are infinite pairs of prime numbers whose distance is less than a certain value.
But for the mathematical community, whether it is the weak Goldbach conjecture or the weak twin prime number theorem, they are just a prelude to climbing to the top.
They are like a loud national anthem for a climber who climbs Mount Everest before setting off. They can give the climber courage to a certain extent, but it is unrealistic to expect to climb Mount Everest and stand on the top.
"Xu, will you try to develop in the direction of number theory?"
After the atmosphere was slightly silent, 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 will have the opportunity to achieve a huge fruit in the field of prime numbers?
He didn't dare to say for sure, after all, who could 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 the rising star in mathematics in front of him to plunge into it for years or even decades without making any 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 were never solved.
There is even no hope of solution.
However, if Xu Chuan continues to study spectral theory, functional analysis, and Dirichlet function, I cannot say that he will definitely be able to make a greater contribution than the Weyl-Berry conjecture, but he will definitely be able to further expand the boundaries and expand the scope of these fields. mathematical scope.
But when it comes to number theory, it becomes uncertain.
Not every genius is Tao Zhexuan. At present, Xu Chuan's mathematical talent is indeed higher than Tao Zhexuan, but no one knows what will happen after crossing fields.
Xu Chuan did not give Avila a definite answer. In the past year, he did read a lot of books related to number theory, but number theory was not in his subsequent study and research arrangements.
He prefers functions and analysis that can be applied in practical applications to solve physical problems, while number theory mainly studies the properties of integers, which is considered pure mathematics.
Of course, with the development of mathematics today, it cannot be said that any field of mathematics is pure mathematics. It can always be linked to other fields.
For example, in statistical mechanics, the partition function is the basic mathematical object of study; and in the analytical theory of prime number distribution, the zeta function is the basic object.
This unorthodox interpretation of the zeta function as a partition function therefore points to a possible fundamental connection between the distribution of prime numbers and this branch of physics.
However, at present, there is still a vacancy in applying number theory to the field of physics, which is far less extensive than mathematical analysis, function transformation, and mathematical models.
Therefore, Xu Chuan is not very inclined to invest a lot of energy and time in the field of pure number theory.
But it is definitely a good idea to study and learn number theory.
Because number theory is not just pure number theory, but also various branches such as analytic number theory, algebraic number theory, geometric number theory, computational number theory, arithmetic algebraic geometry, etc.
These branches are all extended from pure number theory, that is, elementary number theory, combined with other mathematics.
For example, analytic number theory uses calculus and complex analysis (ie, complex functions) to study number theory about integer problems.
The things he talked about with Professor Avila tonight have something to do with analytic number theory,
Because in addition to the circle method, sieve method, etc., analytical number theory methods also include modular form theory related to elliptic curves, etc. Later, it developed into the theory of automorphic forms, which was connected with the representation theory.
Therefore, having a certain foundation in number theory will be of great help to other mathematics learning.
(End of chapter)