The charm of mathematics is that the difficulty of learning various knowledge theories will not change due to the passage of time. Instead, it will become more and more mellow and fascinating as time goes by...
pain.
The mathematical theories are all there.
Quietly, neither sad nor happy, every mathematical theory contains information entropy that ordinary people cannot comprehend and absorb in a lifetime.
The abyss of mathematics is composed of these profound and extremely difficult to understand mathematical theories.
This is a test that every wise man must go through on the road ahead. There is no way to avoid it, there are no shortcuts, and opportunism and overtaking in corners are meaningless in front of it.
As for the question of whether there is a "bottom" to the mathematical abyss, based on the existing knowledge system and cognition, the answer is naturally - no.
Analytic number theory, a cutting-edge branch of basic mathematics, uses mathematical analysis as a research tool. The number theory school uses mathematical analysis to study the properties of integers and prove them. It originated from the Goldbach conjecture, twin prime conjecture, prime number distribution, Waring problem and lattice problem.
The research is one of the contents of advanced number theory.
In the cognition of most ordinary people, their understanding of the subject of analytic number theory basically comes from the well-known Goldbach's conjecture.
Of course, even Goldbach's conjecture cannot be understood in a true sense by many people. More than half of the people understand it as proving that '1+1=2'.
Analytical number theory mainly studies the distribution of prime numbers, which are called prime numbers. At this time, a conventional question will arise - why study prime numbers?
Because it's fun.
Okay, that's bullshit.
Because prime numbers with unique properties play a great role in the study of number theory. All positive integers greater than 1 can be expressed as the sum of prime numbers. In a sense, the status of prime numbers in number theory is similar to that in physics.
If we can master the secret of the distribution of prime numbers, the atoms that make up all things among them, we may be able to master the secret of the universe.
You must know that the definition of prime numbers is so simple that even elementary school students can understand it, but the distribution of prime numbers is extremely mysterious and weird, and it is completely unconventional. This is an important reason why there are countless folk studies to prove Goldbach's conjecture in later generations.
Finding one prime number is easy, finding ten prime numbers is easy, but finding the distribution pattern of all prime numbers is extremely difficult.
From Euler to Gauss, and from Gauss to Riemann, the law of prime number distribution is still not fully grasped.
When it comes to analytic number theory, we have to mention that the founder of the entire analytic number theory school comes from the god-level apostle and founder of mathematics - Bornhard Riemann.
The founder Riemann pioneered the establishment of analytic number theory based on the mathematical emperor Euler and his mentor Gauss, which opened a new ceiling for the number theory research at that time and also provided an iron rice bowl for the current Yu Hua.
Apart from anything else, at least with his research on analytic number theory, it would not be a problem to work as a teaching assistant or lecturer in any university across the country.
Thanks to the ancestor for giving me a bite of rice.
Yu Hua, who was studying the whole book, read the wonderful parts and sighed silently in his heart.
In terms of seniority, Riemann is his ancestor.
In terms of achievements, the founder is completely in the god-level echelon composed of Euler, Newton and Gauss.
In terms of mathematical contributions, Riemann almost single-handedly established the system of modern mathematics. He was not only the father of analytic number theory, but also the father of complex variable function theory and Riemannian geometry. He also pioneered combinatorial topology and algebraic geometry.
sexual contribution.
On talent...
Now, he seems to be on par with the Patriarch, at least, he can see the Patriarch's back.
The former him, sorry to bother you...
Analytical number theory is mainly studied through Euler constants, complex functions, circle method, sieve method, exponential sum method, characteristic sum method, density and other methods.
The introductory introduction to number theory in my hand mainly records Master Hua Luogeng’s research on number theory. According to the catalog, the content of the notes consists of elementary number theory and advanced number theory. Most of them are basic knowledge, and a few are advanced knowledge points, such as the geometry of numbers and the prime number theorem.
, Overview of the distribution of prime numbers, as well as the classic Waring problem and Goldbach's conjecture.
It has to be said that Master Hua Luogeng is the first person in China to study Goldbach's conjecture. Although there is no proof of '5+5' in it, the content of the notes has completely proved this year's Italian Professor Lacey's new approach to Goldbach's conjecture.
Results.
It is very difficult to prove '5+5', and it is difficult to completely prove the existing results such as '5+7' and '4+9'.
The whole proof process has benefited Yu Hua a lot, and the part of the notes about Goldbach's conjecture, in addition to Professor Lacey's latest results this year, also includes previous achievements such as '6+6' and '7+7'
complete proof.
Among them, the sieve method of analytical number theory plays an irreplaceable role.
Regarding the famous Hualin problem, Yu Hua only saw the content of the problem and the deduction of some basic properties, but failed to see the famous Wahler's theorem and Wahler's inequality in history.
However, this is normal. First, Fahrenheit's Theorem was published in 1940, and it is now 1937, which is still more than two years away. Second, such an important mathematical result would not be listed here even if it existed.
This is the mathematical result of a mathematician's painstaking efforts, and it is of extremely important importance.
Regarding this point, Yu Hua said that he could understand it and didn't have any small thoughts.
There are many scientific achievements that can maintain the character of a genius. Yu Hua does not need to worry about Fahrenheit's theorem. Besides, he cannot do anything to steal his master's achievements.
Some scientific results can be taken freely.
Some scientific results cannot be obtained.
‘Some’ depends on nationality, identity, contribution to the country, importance of results, mood and many other factors.
"I have already understood the basic concepts of analytic number theory. It turns out that studying prime numbers is so interesting. No wonder so many mathematicians never stop pursuing prime numbers..." Yu Hua, who stayed in the master bedroom, carefully read the complete introduction to number theory. He was very interested in analysis.
I have gained a comprehensive and systematic understanding of number theory and various number theory issues, and my knowledge level has improved again. I closed my eyes, carefully recalled and understood the information entropy contained in each mathematical formula and method.
Understand, digest.
It's like eating.
Since the interview for the entrance examination for recommended students at Tsinghua University, Yu Hua has been working on Yu's Seven Pagodas while studying Introduction to Number Theory. However, the mathematical knowledge contained in it is extremely difficult, so Yu Hua's learning progress is not fast.
Although, these things basically belong to the "basics" of advanced number theory.
After so many days, I finally digested the entire book.
After reading Master Hua Luogeng's book, Yu Hua basically got an introduction to analytic number theory, and truly stepped into the path of no return in the study of prime numbers.
interesting.
Really interesting.
Although Yu Hua has always regarded mathematics as a tool, this in no way hinders the fun of studying prime numbers.
"With my current academic level and strength, I can only say that I have touched some thresholds, which is not enough to prove '5+5'. Also, I may need to add some academic achievements as a transition. After all, I am only a quasi-college student.
For an achievement that is not an achievement, the number should be set at two."
After consuming all the knowledge content, Yu Hua opened his eyes and changed from the academic research state to the normal thinking state: "Asymmetric cryptography can be used as a transitional result. As for the second transitional result, it is best to have a close relationship with prime number research and have
An important influence, enough to elevate my identity and academic status to a higher level..."
After thinking carefully in my mind, my eyes inadvertently looked at the table of contents of my desktop notes - The Prime Number Theorem.