The Science Fiction World of Xueba

In general, conjectures in the field of number theory are more precise and intuitive to express.

For example, Fermat's last 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 about x, y, and z has no positive integers untie.

Another example is the famous Goldbach's 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's very abstract to understand.

Simply put, there are 3 numbers: a, b, and c = a+b. If these 3 numbers are relatively prime and there is no common factor greater than 1, then multiply the prime factors of these 3 numbers together to get d, it seems, will usually be larger than c.

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

These 3 numbers are relatively prime, so the multiplication of non-repeating factors will result in d=2*7*3=42 being greater than c=9.

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

However, this is only a law that seems to be correct, and there are actually counterexamples!

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

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

We can then state the ABC conjecture that d is "usually" not "much smaller" than c.

How do you say that it is usually not much smaller than c?

If we enlarge d a little bit to d (1+ε power), then although it is still not guaranteed to be larger than c, it is enough to make the number of counterexamples from infinite to finite.

This is the expression of the ABC conjecture.

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

Therefore, the difficulty of this conjecture can be imagined.

In fact, apart from the unresolved crown Riemann conjecture involving multiple branches of mathematics, other conjectures in number theory, such as Goldbach's conjecture, the twin prime conjecture, and the solved Fermat's last theorem, basically have no ABC conjecture is important.

Why?

First, the ABC conjecture is counterintuitive to number theory researchers.

There are countless counter-intuitive but proven correct theories in history.

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

To give a simple example: the law of inertia of Newtonian mechanics, the object will maintain the current state of motion unless there is no external force, which was undoubtedly a heavyweight thought bomb in the 17th century.

Objects will of course change from motion to stagnation when there is no force. This is the normal thinking of ordinary people at that time based on their daily experience.

In fact, this kind of thinking will seem 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 people at that time, the theorem of inertia was indeed quite contrary to human common sense!

The ABC conjecture is to current number theory researchers what Newton's law of inertia was to ordinary people in the seventeenth century, and it violates common sense in mathematics.

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

One reason is that allowing addition and multiplication to interact algebraically creates infinitely possible and unsolvable problems,

For example, Hilbert's tenth problem about the unified methodology of the Diophantine equation has long been proved to be impossible.

If the ABC conjecture is proved to be correct, then there must be a mysterious relationship between addition, multiplication and prime numbers that has never been touched by human known mathematical theories.

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

For example, the Diophantine equation problem just mentioned, the extension 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 proved, such as Fermat's last 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 proven, it will have a huge impact on number theory, which is no different from relativity and quantum physics to modern physics.

Because of this, when Mochizuki Shinichi claimed to have proved the ABC conjecture in 2012, it caused such a big stir in the mathematics world.

Shinichi Mochizuki was born in Tokyo, Japan on March 29, 1969. At the age of 16, he entered Princeton University in the United States for undergraduate studies. Three years later, he entered graduate school and studied under the famous German mathematician and winner of the Fields Medal in 1986. (ie 1992) received a Ph.D. in Mathematics.

Even in the eyes of Faltings, who has always been strict and poisonous, Mochizuki Shinichi can be regarded as one of his favorite students.

In 1992, Mochizuki returned to Japan as a researcher at the Institute of Mathematical Analysis at Kyoto University because of his eccentric personality and his inability to adapt to American culture.

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

After 1998, Mochizuki devoted all his energy to the proof of the ABC conjecture, and almost disappeared from the mathematical community.

It was not until 2012 that Shinichi Mochizuki published a 512-page proof paper on the ABC conjecture that it once again aroused large-scale attention in the mathematics community.

To some extent, Shinichi Mochizuki is somewhat similar to Perelman, except that Perelman successfully proved the Poincaré conjecture, while Shinichi Mochizuki’s proof of the ABC conjecture has not been recognized by the mathematics community.

Mochizuki's theoretical tool for studying the ABC conjecture is Far Abelian geometry.

Therefore, before studying Mochizuki's ABC conjecture paper, Pang Xuelin also asked Tian Mu to find Mochizuki's related works on Far Abelian geometry.

Founded in the 1980s by Pope Grothendieck of algebraic geometry, Far Abelian geometry is a very young discipline in mathematics.

The subject of this discipline is the structural similarity of fundamental groups of algebraic varieties on different geometric objects.

Banach, the father of modern analysis, said: "A mathematician can find the similarities between theorems, a good mathematician can see the similarities between the proofs, and an excellent mathematician can perceive the similarities between branches of mathematics." similarities. Finally, the research mathematician can overlook the similarities between these similarities."

Grothendieck can be regarded as a real research mathematician, and Far Abelian geometry is a branch of mathematics that studies "similarity of similarity".

From the Italian mathematicians Ferro and Tartaglia in the 16th century who discovered the formula for finding the roots of cubic equations in one variable (that is, the Cardano equation), to the discovery of the group structure of solutions to special higher-order equations by Galois in the 19th century.

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

The fundamental group of algebraic varieties is a synthesis of the theory of algebraic varieties that has synthesized a large class of theories. It is concerned with what kind of structure is independent of the representation of algebraic varieties of geometric objects.

Therefore, for mathematicians, another difficult problem to check whether Mochizuki's proof has errors or omissions is: to thoroughly understand Mochizuki's 512-page ABC conjecture proof, you need to understand Mochizuki's far Abelian geometry. 750 pages of work!

In total, only about 50 mathematicians in the world have enough background knowledge in this area to read Mochizuki Shinichi's book on far Abelian geometry, not to mention the "generalized Teichmüller theory" established by Mochizuki in proving conjectures.

So far, this theory has only been understood by Mochizuki himself.

Pang Xuelin didn't expect to be able to study the ABC conjecture thoroughly in just a few years. He just wanted to use his time on Mars to understand the relevant ideas of Mochizuki Shinichi's research on the ABC conjecture, and to find the mistakes and omissions in the paper. place.

Of course, if you can get some inspiration from it, that would be great.