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

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

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.

To put it simply, there are 3 numbers: a, b, and c = a+b, and if these 3 numbers are coprime and there is no common factor greater than 1, then multiplying the quality factors of these 3 numbers without repeating the quality factor to obtain d will seem to be usually larger than c.

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

These 3 numbers are coprime, so the multiplication of the non-repeating factors has d=2*7*3=42>c=9.

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

However, this is just 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 distributed computing, a distributed computing platform based on BOINC, to find counterexamples of the ABC conjecture, one of which is 3+125=128: where 125=5^3 and 128=2^7, then the multiplication of the non-repeating quality factor is 3*5*2=30, and 128 is larger than 30.

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

So we can formulate ABC's conjecture that d is "usually" no less than c "usually".

How can it be called usually not much smaller than c?

If we amplify d a little bit to the (1+ε power) of d, then it is still not guaranteed to be greater than c, but it is enough to change the counterexamples from infinite to finite.

This is the formulation 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 a bit of a power (1+ε power) in a vague way, and the most unfortunate thing is that there are counterexamples.

Therefore, the difficulty of this conjecture can be imagined.

In fact, with the exception of the unsolved conjecture Crown Riemann conjecture involving multiple branches of mathematics, other conjectures in number theory, such as the Goldbach conjecture, the twin prime conjecture, and the solved Fermat's theorem, are basically not as important as the ABC conjecture.

Why is that?

First of all, the ABC conjecture is counterintuitive for number theorists.

There are countless theories that have historically been counter-intuitive but have been proven to be correct.

Once counter-intuitive theories were proven correct, they basically changed the course of scientific development.

To give a simple example: Newtonian mechanics' law of inertia, in which an object would remain in its current state if it was not exposed to an external force, was undoubtedly a heavyweight thought bomb in the 17th century.

Of course, when an object is not forced, it will change from motion to stop, which is the normal thought of ordinary people at that time based on daily experience.

In fact, this idea would seem too naïve to anyone who studied junior high school physics in the 20th century and knew that there is a force called friction.

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

The ABC conjecture is to today's number theorists what Newton's law of inertia was to ordinary people in the seventeenth century, and it is even more contrary to mathematical common sense.

This common sense is: "The quality factors of A and B should not have any connection with the quality factors of their sum." ”

One reason for this is that allowing addition and multiplication to interact algebraically creates infinite possibilities and unsolvable problems, such as Hilbert's tenth problem on the unified methodology of the Diophantine equation, which has long been proven impossible.

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

Moreover, the ABC conjecture has a significant connection with many other unsolved problems in number theory.

For example, the Diophantine equation problem mentioned earlier, the generalized conjecture of Fermat's final theorem, the Mordell conjecture, the Erdős–Woods conjecture, and so on.

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

From this point of view, the ABC conjecture is a powerful probe of the unknown universe with a prime structure, second only to the Riemann conjecture.

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

Because of this, Shinichi Mochizuki caused such a stir in the mathematical community in 2012 when he claimed to have proved the ABC conjecture.

Shinichi Mochizuki was born on March 29, 1969 in Tokyo, Japan, and entered Princeton University in the United States at the age of 16 for undergraduate studies, and entered graduate school three years later, under the tutelage of the famous German mathematician, Faltins, winner of the 1986 Fields Medal, and received a doctorate in mathematics at the age of 23 (that is, in 1992).

Even in the eyes of the always strict and venomous Faltins, Shinichi Mochizuki was one of his protégés.

In 1992, because of his withdrawn and eccentric personality and not adapting to American culture, Shinichi Mochizuki returned to Japan as a researcher at the Institute of Mathematical Analysis at Kyoto University.

During this period, Shinichi Mochizuki made outstanding contributions to the field of "Far Abel Geometry", for which he was invited to give a one-hour lecture at the International Congress of Mathematicians in Berlin in 1998.

After 1998, Mochizuki began to devote all his energy to the proof of the ABC conjecture, and almost disappeared from the mathematical world.

It wasn't until 2012 that Shinichi Mochizuki published a 512-page paper on the proof of the ABC conjecture, which once again attracted massive attention from the mathematical community.

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

Shinichi Mochizuki's theoretical tool for studying the ABC conjecture is far-abelian geometry.

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

Far Abelian geometry, created by Pope Grothendieck of algebraic geometry in the eighties of the twentieth century, is a very young discipline in mathematics.

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

Barnach, the father of modern analytics, said, "Mathematicians can find similarities between theorems, good mathematicians can see similarities between proofs, and great mathematicians can perceive similarities between branches of mathematics." Finally, the mathematician can overlook the similarities between these similarities. ”

Grothendieck is a true mathematician, and far-abelian geometry is a branch of mathematics that studies the "similarity of similarity".

From the 16th century Italian mathematicians Ferro and Tartaglia discovered the formula for finding the root of a univariate cubic equation (i.e., the Cardano equation), to the 19th century Galois discovered the group structure of a special higher order equation solution.

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

The basic group of algebraic clusters is a synthesis of algebraic cluster theory, which has already synthesized a large class of theories, and is concerned with what kind of structure is independent of the appearance of algebraic clusters of geometric objects.

Therefore, another difficult problem for mathematicians to check for errors and omissions in Mochizuki Shinichi's proof is that in order to fully understand the proof of Mochizuki's 512-page ABC conjecture, it is necessary to first understand Shinichi Mochizuki's 750-page work on far-abelian geometry!

In total, there are only about 50 mathematicians in the world who have enough background knowledge to read through Shinichi Mochizuki's work on far Abel geometry, not to mention the "generalized Tayhimüller theory" that Mochizuki established in his proof conjecture.

So far, this theory has only been figured out by Shinichi Mochizuki himself.

Pang Xuelin didn't expect that he would be able to study the ABC conjecture thoroughly in just a few years, he just wanted to use his years on Mars to figure out the relevant ideas of Shinichi Mochizuki to study the ABC conjecture and find the errors and omissions in the paper.

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