The Science Fiction World of Xueba

In general, the conjectures in the field of number theory are expressed more accurately and intuitively.

For example, Fermat's Great Theorem, which has been proved by Andrew Wiles, can be directly expressed as: when the integer n is greater than 2, the equation x ^ n + y ^ n = z ^ n has no positive integer. solution.

比如 Another example is the famous Goldbach conjecture, which can be understood in one sentence: any even number greater than 2 can be written as the sum of two prime numbers.

But the ABC conjecture is an exception.

It is very abstract to understand.

Simply put, there are three numbers: a, b, and c = a + b. If these three numbers are prime, and there is no common factor greater than 1, then the three prime numbers that are not repeated are multiplied. d, seems to be usually larger than c.

For example: a = 2, b = 7, c = a + b = 9 = 3 * 3.

These three numbers are coprime, so multiplying the non-repeating factors has d = 2 * 7 * 3 = 42 greater than c = 9.

You can also experiment with several sets of numbers, such as: 3 + 7 = 10, 4 + 11 = 15, which also meet this seemingly correct rule.

But this is just the law that looks right, there are actually counter-examples!

The ABC @ home website operated by the Institute of Mathematics at the University of Leiden in the Netherlands is using the BOINC-based distributed computing platform to find counterexamples of the ABC conjecture. One counterexample is 3 + 125 = 128: 125 = 5 ^ 3, 128 = 2 ^ 7, then the multiplication of non-repeating prime factors is 3 * 5 * 2 = 30, 128 is greater than 30.

In fact, computers can find an infinite number of such counterexamples.

Then we can express the ABC conjecture that d is "usually" not "too much smaller" than c.

How can I say that it is usually not much smaller than c?

If we magnify d a little bit to d's (1 + ε power), then although it is still not guaranteed to be larger than c, it is enough to make the counterexample change from infinite to finite.

This is the expression of the ABC conjecture.

ABC conjecture involves not only addition (sum of two numbers), but also multiplication (multiplication of prime factors), and then it also has vague point powers (1 + ε power). The worst thing is that there are counterexamples.

Therefore, the difficulty of this conjecture can be imagined.

In fact, in addition to the unresolved Riemann conjecture in the world of conjecture involving multiple branches of mathematics, other conjectures in number theory, such as the Goldbach conjecture, the twin prime conjecture, and the Fermat theorem that have been solved, basically have no The ABC guess is important.

Why is this?

First, the ABC conjecture is counter-intuitive for number theory researchers.

反 Countless counter-intuitive theories have been proven to be correct in history.

Once the counter-intuition theory was proved correct, it basically changed the course of scientific development.

To give a simple example: the law of inertia of Newtonian mechanics, if the object is not subject to external forces, it will maintain its current state of motion, which is undoubtedly a heavyweight thought bomb in the 17th century.

Of course, when an object is under no force, it will change from motion to stop. This is the normal thought of ordinary people at that time based on daily experience.

实际上 In fact, this kind of thinking will look too naive to anyone who has studied junior high school physics in the 20th century and knows that there is a force called friction.

But for the people at the time, the theorem of inertia was indeed quite contrary to human common sense!

ABC's conjecture to the current number theory researchers is just like the Newton's law of inertia of ordinary people in the 17th century, which is a violation of mathematical common sense.

This common sense is: "The prime factors of a and b should not have any relationship with the prime factors of their sum."

One of the reasons is that allowing addition and multiplication to interact in algebra will produce infinitely possible and unsolvable problems, such as Hilbert's tenth problem about the unified methodology of Diophantine equations, which has long been proved impossible.

If the ABC conjecture proves to be correct, then there must be a mysterious connection between the mathematical theory known to mankind that addition, multiplication, and prime numbers have never touched.

Moreover, the ABC conjecture has important connections with many other unsolved problems in number theory.

For example, the Diophantine equation problem just mentioned, the generalized conjecture of Fermat's last theorem, the Mordell conjecture, the Erd? S–Woods conjecture, and so on.

Moreover, the ABC conjecture can also indirectly deduce many important results that have been proven, such as Fermat's final theorem.

From this perspective, the ABC conjecture is a powerful detector of the unknown universe with a prime number structure, second only to the Riemann conjecture.

Once the ABC conjecture is proved, the impact on number theory is as great as that of relativity and quantum physics to modern physics.

Because of this, when Mochizuki claimed in 2012 that she proved the ABC conjecture, it would cause such a sensation in the mathematical world.

Mochizuki Shinichi was born in Tokyo, Japan on March 29, 1969. At the age of 16, he entered the undergraduate program at Princeton University in the United States and entered the graduate school three years later. He studied under the famous German mathematician. (Ie 1992) PhD in Mathematics.

Even in the eyes of Faltins, who has always been strict and poisonous, Mochizuki Shinichi is one of his proud students.

In 1992, because of his relatively eccentric personality, he did not adapt to American culture. Mochizuki returned to Japan as a researcher at the Institute of Mathematical Analysis of Kyoto University.

During this period, Mochizuki Shinichi made an outstanding contribution in the field of "Far Abelian Geometry", and was therefore invited to give an hour-long speech at the 1998 International Congress of Mathematicians in Berlin.

After 1998, Mochizuki started to devote all her energy to the proof of the ABC conjecture, and almost disappeared in the mathematical world.

It wasn't until 2012 that Mochizuki published a 512-page ABC conjecture proof paper, which once again aroused large-scale attention in the mathematical community.

To some extent, Mochizuki Shinichi is somewhat similar to Perelman, but Perelman successfully proved Poincaré's conjecture, while Mochizuki Shinichi's ABC conjecture proved that it has not been recognized by the mathematical community.

Wang Yueyue's new theoretical tool for studying ABC conjecture is far Abelian geometry.

Therefore, before studying the new ABC conjecture of Mochizuki, Pang Xuelin asked Tian Mu to find Mochizuki's new works on Far Abelian geometry.

The Yuanyuan Abelian geometry was created by Pope Grothendieck of the algebraic geometry in the 1980s and is a very young discipline in mathematics.

对象 The research object of this discipline is the structural similarity of basic groups of algebraic clusters on different geometric objects.

"The mathematicians can find similarities between theorems," said Barnach, the father of modern analytics. "UU reading www.uukanshu.com. Excellent mathematicians can see the similarities between proofs. Excellent mathematicians can Perceive similarities between branches of mathematics. Finally, advanced mathematicians can overlook the similarities between these similarities. "

Grothendieck is a real mathematician in the real sense, and Far Abelian geometry is a branch of mathematics that studies "similarities and similarities."

From the sixteenth century Italian mathematicians Ferro and Tartaglia discovered the root-seeking formula of the one-ary cubic equation (the Cardano equation), to the nineteenth century Galois discovered the group structure of the special higher-order equation.

Algebraic clusters in unitary algebraic geometry are common solutions to a large class of equations.

The basic group of unitary algebraic clusters is another synthesis of the algebraic cluster theory that has integrated a large class of theories, and cares about what kind of structure is independent of the appearance of algebraic clusters of geometric objects.

Therefore, for mathematicians, another difficulty in checking whether Mochizuki Shinichi's proof is wrong: To fully understand Mochizuki's 512-page ABC conjecture proof, you need to understand Mochizuki's new far-abel geometry. 750 pages of work!

总共 In total, only about 50 mathematicians in the world have enough background knowledge in this area to read through the far-Abelian geometry work of Mochizuki Shinichi, not to mention the "generalized Tichmüller theory" established by Mochizuki in the proof conjecture.

So far, only Mochizuki Shinichi can understand this theory.

Pang Xuelin did not expect that he could thoroughly study the ABC conjecture in just a few years. He just wanted to use his years in Mars to figure out the relevant ideas of Mochizuki's new research on the ABC conjecture and find the mistakes and omissions in the paper. Office.

Of course, if you can get any inspiration from it, it would be better.