162 1 year, Fermat bought a new French translation of Diophantine's Arithmetic in Paris, in which the Pythagorean triangle was discussed. While reading a book, he made a short note to the effect that although the equation X 2+Y 2 = Z 2 has countless integer solutions, the similar equation X N+Y N = Z N will never have a solution when n is greater than 2.

"I found a good way to prove it," Fermat wrote, "but writing it down is too narrow."

-From One to Infinity by george gamow.

1, "Can you compare two infinite numbers which is bigger?" ?

Some numbers are infinite, and no matter how long it takes, they are bigger than the numbers we write down. Obviously, "the number of all numbers" is infinite, and so is "the number of geometric points on a line". Are there any other ways to describe these numbers besides infinity? For example, can you compare two infinite numbers which is bigger?

"Are there more numbers or more dots on a line?" Is such a question meaningful? These interesting questions at first glance were first put forward by the famous mathematician Georg Cantor, who was also the father of "infinite number arithmetic".

2. "Unlimited number of sizes"

To discuss the size of infinite numbers, we must first face a problem, that is, to compare the two numbers we say or write down, which is similar to hottentots knowing how many glass beads or copper coins he has when he looks in a treasure chest. But remember, hottentots can only count to three at most. So since he can't count more, should he give up comparing the number of glass beads with the number of copper coins? Of course not. If he is smart enough, he can compare beads and copper coins one by one and get the answer. He put a bead and a coin together, the second bead and the second coin together, and so on. If the last bead is used up and there are coins left, then he knows that he has more copper coins than glass beads; On the contrary, he has more glass beads; If both are used up at the same time, then he has the same number of two things.

Cantor's method of comparing the sizes of two infinite numbers is exactly the same: if we pair the object sets represented by two infinite numbers, so that every object in one infinite set is paired with an object in another infinite set, and there are no redundant objects in the last two sets, then the infinite numbers representing the two sets are equal. However, if one of the sets has a surplus, then we can say that the infinite number representing this set is greater or stronger than the infinite number representing another set.

"In an infinite world, parts may be equal to the whole."

According to our infinite number comparison rule, we must admit that the number of all even numbers is equal to the number of all numbers. Of course, this sounds ridiculous, because even numbers are only a part of all numbers, but don't forget that we are dealing with infinite numbers here, so we must be prepared for the different characteristics we encounter.

In fact, in an infinite world, "part may be equal to the whole"! A story about the famous German mathematician david hilbert can illustrate this point well. It is said that in his lecture on infinity, he used the following words to illustrate the contradictory characteristics of infinity:

"Let's imagine a hostel with a limited number of rooms and assume that all the rooms are full. Then a new guest came and wanted to book a room. "I'm sorry," the boss will say, "but it's full." Now let's imagine a hotel with countless rooms, all of which are full, and then a new guest comes and wants to book a room.

"'of course!' The boss shouted, and then he moved the people in room 1 to room 2, room 2 to room 3, room 3 to room 4, and so on. Then, after this transfer, 1 room became vacant and new tenants moved in.

"Let's imagine a hostel with countless rooms, all of which are full. At this time, unlimited new guests come to make reservations.

"'Well, gentlemen,' said the boss,' don't be impatient.'

"He moved the guests in room 1 to room 2, room 2 to room 4, room 3 to room 6, and so on.

"Now that the odd rooms are vacant, it is easy to accommodate an unlimited number of new guests."

Because it is wartime, even in Washington, the situation described by Hilbert is difficult to understand, but this example vividly describes the characteristics of infinite numbers, which is completely different from what we usually encounter in arithmetic.

4. "Hilbert: There is no similarity between pure mathematics and applied mathematics, and there is no comparability at all."

Mathematics is usually regarded as the queen of science by people, especially mathematicians, and as a queen, she will naturally try to avoid being subordinate to other disciplines. For example, Hilbert was invited to give a public speech at a joint meeting of pure mathematics and applied mathematics, breaking the hostility between the two mathematicians. This is what he said:

"People often say that pure mathematics and applied mathematics are relative. This sentence is wrong. Pure mathematics and applied mathematics are not opposed to each other. They have never been against each other before and will never be against each other in the future. This is because there is no similarity between pure mathematics and applied mathematics, and there is no comparability at all. "

5. "Most theorems in number theory are conceived when people deal with different numerical problems, just as laws in physics are the result of people dealing with problems related to physical objects."

Although mathematicians want to keep the purity of mathematics and are insensitive to other disciplines, other disciplines, especially physics, favor mathematics and try their best to establish a "friendly relationship" with it. In fact, almost every branch of pure mathematics is now used to explain this or that feature in the physical universe. These subjects, including abstract group theory, noncommutative algebra and non-Euclidean geometry, have always been considered absolutely pure and will not have any practicality.

However, so far, there is still a big system in mathematics that has no practical application except training thinking, and can simply be crowned as the "pure crown" of honor. This is the so-called "number theory" (here refers to integers), one of the oldest branches of mathematics and one of the most complicated products of pure mathematical thinking.

Incredibly, as the purest part of mathematics, number theory can be called an empirical science or even an experimental science in a sense. In fact, most theorems in number theory are conceived when people deal with different numerical problems, just as laws in physics are the result of people dealing with problems related to physical objects. And like physics, some theorems in number theory have been proved "from a mathematical point of view", while others are still in a purely empirical stage, challenging the brains of the best mathematicians.

6. Is the number of prime numbers infinite, or is there a maximum prime number? "

Take the prime number problem as an example. The so-called prime number is a number that cannot be expressed by the product of two or more smaller numbers. Numbers like 1, 2,3,5,7 are prime numbers, but 12 is not, because 12 can be written as 2×2×3.

Is there an infinite number of prime numbers, or is there a maximum prime number, and all numbers greater than it can be expressed by the product of several known prime numbers? This problem was first proposed and studied by Euclid. He gave a concise and clear demonstration method and proved that the number of prime numbers is infinite, so there is no so-called "maximum prime number".

In order to verify this problem, we assume that the number of all known prime numbers is limited, and the letter n is used to represent the largest known prime number. Now let's calculate the product of all known prime numbers, plus 1, which is expressed by the following formula:

( 1×2×3×5×7× 1 1× 13×…×N)+ 1

Of course, this number is much larger than the maximum prime number n we proposed, but it is obviously impossible for this number to be divisible by any known prime number (including n at most), because from its structure, if this number is divided by any other prime number, it will leave a remainder 1.

Therefore, this number is either a prime number itself, or it must be divisible by a prime number greater than n, but both cases contradict our initial assumption that "n is the largest known prime number".

This test method, called reduction to absurdity, is one of the favorite methods used by mathematicians.

7. "Is there a simple way to write down all the prime numbers one by one?"

Now that we know that the number of prime numbers is infinite, we have to ask ourselves, is there a simple way to write down all the prime numbers one by one? Eratosthenes, an ancient Greek philosopher and mathematician, first proposed a method called "Eratosthenes Screening Method". You just need to write down the complete integer sequence, 1, 2, 3, 4, etc. , and then delete all multiples of 2, then delete all multiples of 3, multiples of 5 and so on. The first 100 integers, including 26 prime numbers, were screened by Eratostheny screening method. By using this simple screening method, we get all the prime numbers within 654.38+0 billion.

However, if we can extract a formula that can only calculate prime numbers and calculate all prime numbers quickly and automatically, it will be simpler. However, after centuries of efforts, people still have not got such a formula. 1640, the famous French mathematician Fermat thought he had deduced a formula that could only calculate prime numbers.

8. "Goldbach conjecture: any even number can be expressed as the sum of two prime numbers"

There is still an interesting theory in number theory that has not been proved or overthrown so far. This is the Goldbach conjecture put forward by Goldbach 1742, which claims: "Any even number can be expressed as the sum of two prime numbers." Take some simple figures as examples, and you can easily find that this sentence is correct, such as 12 = 7+5, 24 = 17+7, 32 = 29+3. Although mathematicians have done a lot of work on this issue, they still can't give a decisive evidence to prove that this statement is absolutely correct, and they can't find a counterexample to prove that it is wrong. 193 1 year, the Soviet mathematician Schnuel ellman took a crucial step towards decisive evidence. He successfully proved that "any even number can be expressed as the sum of no more than 300,000 prime numbers". Later, the gap between "the sum of 300,000 prime numbers" and "the sum of two prime numbers" was greatly narrowed by another person, vinogradov, who simplified the former to "the sum of four prime numbers". However, the last two steps from Victor nogueira's four prime numbers to Goldbach's two prime numbers seem to be the most difficult, and no one can be sure how many years or centuries it will take to confirm or overturn this difficult proposition.

9. "The prime ratio between 1 and any number n greater than1is approximately equal to the natural logarithm of n"

Well, it seems that we still have a long way to go to derive a formula that can automatically calculate all arbitrarily large prime numbers, not to mention that we can't guarantee that such a formula will exist.

We can ask a slightly simpler question-about the proportion of prime numbers in a given numerical interval. Will this ratio remain the same as the number gets bigger? If so, will it increase or decrease?

? Is there a simple way to describe this ratio that decreases with the increase of numerical value? Not only that, the average distribution of prime numbers is one of the most remarkable discoveries in the whole mathematical field. Simply put, "the ratio of prime numbers from 1 to any number n greater than 1 is approximately equal to the natural logarithm of n". The greater the n, the closer the two values are.

? Like many other theories in number theory, the above prime number theory was originally put forward from an empirical point of view, and it could not be proved by strict mathematical methods for a long time later. It was not until the end of 19 that French mathematician Adama and Belgian mathematician Delavallee Posen finally proved this point in an extremely complicated way. It is difficult to explain it clearly in a few words, so I won't go into details here.

10, "Fermat's last theorem"

Since we discuss integers, we have to mention the famous Fermat's Last Theorem, which can be used as an example to discuss problems unrelated to the characteristics of prime numbers. The root of this problem dates back to ancient Egypt, when all excellent carpenters knew that a triangle with a side length ratio of 3: 4: 5 must have a right angle. They used this triangle, which is now called the Egyptian triangle, as their square (in the geometry class in primary school, the Pythagorean theorem was expressed like this: 3 2+4 2 = 5 2).

In the 3rd century, Diophantine began to doubt whether the sum of squares of other two integers was equal to the square of the third number except 3 and 4. He did find some (in fact, countless) number triples with this property, and gave the basic rules for finding these numbers. This right triangle whose three sides are integers is now called the Pythagorean triangle, and the Egyptian triangle is one of them. The construction of a pythagorean triangle can be simply regarded as an equation, in which x, y and z must be integers: x 2+y 2 = z 2.

1 1, "I found a wonderful proof method, but it's too narrow to write it down here."

162 1 year, Fermat bought a new French translation of Diophantine's Arithmetic in Paris, in which the Pythagorean triangle was discussed. While reading a book, he made a short note to the effect that although the equation X 2+Y 2 = Z 2 has countless integer solutions, the similar equation X N+Y N = Z N will never have a solution when n is greater than 2.

"I found a good way to prove it," Fermat wrote, "but writing it down is too narrow."

12, "The most outstanding mathematicians in the world are trying to reproduce the proof method mentioned by Fermat in his notes."

After Fermat's death, people found this book of Diophantine in his reference room, and the notes in the margin were published. That was three centuries ago. Since then, the most outstanding mathematicians in the world have tried to reproduce the proof method mentioned by Fermat in his notes, but there is still no conclusion. But there is no doubt that people have made great progress towards this ultimate goal. At the same time, in the process of trying to prove Fermat's theory, a brand-new branch of mathematics called "ideal number theory" was born. Euler proved that the equations X 3+Y 3 = Z 3 and X 4+Y 4 = Z 4 cannot have integer solutions, and Dirichlet proved that the equations X 5+Y 5 = Z 5 also have no integer solutions. Later, through the joint efforts of several mathematicians, it has been proved that when n is less than 269, the cost will be reduced. But so far, there is still no summary argument to prove that the conclusion is true when the index n takes any value. More and more people suspect that Fermat himself has no proof method, or that he has made a mistake. Later, someone offered a reward of 6.5438 million marks to find the answer, which became a hot topic. Of course, amateurs who only seek money have made no progress.

note:

Fermat's Last Theorem was finally thoroughly proved by British mathematician andrew wiles in 1995. This is a very wonderful story. See Simon Singh's Fermat's Last Theorem: a mystery that has puzzled wise people in the world for 358 years.