Pick the crown jewel
--Goldbach's conjecture
The queen of natural science is mathematics, and the crown of mathematics is number theory. And Goldbach's conjecture is the shining pearl in the crown. Since Goldbach proposed this conjecture in the mid-eighteenth century, countless mathematicians have been attracted by the dazzling brilliance of this pearl and have joined the ranks of picking it. However
No one can succeed.
Eighteenth centuries passed and no one could prove it.
Nineteenth century passed and still no one could prove it.
History has entered the twentieth century, the development of natural science is changing with each passing day, and countless scientific fortresses have been conquered by scientists one by one.
By the 1920s, Goldbach's conjecture began to make some progress. Mathematicians from various countries advanced in a roundabout way, gradually narrowing the encirclement. In this world-wide competition of the century, Chen Jingrun, a well-known Chinese man, defeated mathematics masters from all over the world and won the leading honor. Although Goldbach's conjecture is still just a conjecture, since it was proposed to this day, there are still no other scientific peaks that can obscure its light. History has reached the turn of the century and is about to turn a new page, but mankind still can only enter the 21st century with this regret. Goldbach conjectured, what kind of problem is it?
Find the largest prime number
1, 2, 3, 4, 5,..., these numbers are called positive integers. Among positive integers, numbers that are divisible by 2, such as 2, 4, 6, 8,..., are called even numbers. Those that are not divisible by 2, such as 1, 3, 5, 7,..., are called odd numbers. There is also a number, such as 2, 3, 5, 7, 11, etc., that can only be divided by 1 and itself, but not by other positive integers, it is called a prime number. In addition to 1 and itself, it can also be divided by other positive integers, such as 4,
6, 8, 9, etc., which are called composite numbers. If an integer is divisible by a prime number, the prime number is called the prime factor of the integer. For example, 6 has two prime factors of 2 and 3; and 210 has four prime factors of 2, 3, 5, and 7.
Prime numbers are a very important concept in mathematics. The reason why prime numbers are important was known to the Greek mathematician Euclid as early as 2,000 years ago
. Euclid collected all the mathematical knowledge available to him at the time and wrote a 13-volume mathematical work called "Elements". There is a theorem in the book that is now called the "Fundamental Theorem of Arithmetic": Every natural number greater than
1 is either a prime number or can be expressed as the product of several prime numbers. This expression does not count the prime numbers. The order of the columns is unique.
For example, 630 is the product of 7 prime factors (one of which appears twice):
630=2×3×3×5×7
The part on the right side of the equal sign in the above equation is called the prime factorization of the number 630.
The Fundamental Theorem of Arithmetic tells us that prime numbers are the basic building materials of natural numbers, and all natural numbers are
constructed by them. Prime numbers are much like a chemist's elements or a physicist's elementary particles. By mastering the prime factorization of any number
mathematicians gain almost all information about the number. Therefore, the study of the properties of prime numbers has become one of the oldest and most basic topics in number theory. As early as the time of Euclid, it was proved that there are infinitely many prime numbers. However, to everyone, there seems to be nothing special about prime numbers.
2, 3, 5, 7, 11..., everyone can name a bunch of them casually. But what about the future? Let's
take a look.
We first select a natural number and record it as N; for the number of prime numbers less than N, we record it as π(n
). Comparing the changes in π(n)/n with different values ??of N, we will find that along the sequence of natural numbers, there are fewer and fewer prime numbers.
Table 1: Distribution of prime numbers
N π(n) π(n)/n
10 4 0.400
100 25 0.250
1000 168 0.168
10000 1229 0.123
100000 9592 0.096
1000000 78498 0.078
17th century French mathematician Mersenne proposed a method of finding prime numbers.
In the preface of his book "Cogitata Physica-Mathemati" (Cogitata Physica-Mathemati
c) published in 1644, Mason stated that for n = 2, 3, 5, 7, 13 , 17, 19, 31, 67, 127, 257, the number Mn
=2n-1 is a prime number, and for all other numbers n less than 257, Mn is a composite number. How did he arrive at this conclusion? No one knows. But he did come surprisingly close to the truth. It wasn't until the advent of desktop computers in 1947 that anyone could check his conclusions. He only made 5 mistakes: M67 and M257 are not prime numbers, but M61, M89 and M107 are prime numbers.
Mersenne numbers provide a beautiful way to find very large prime numbers. The function 2n grows rapidly with the increase of n. This ensures that the Mersenne number Mn will soon become extremely large. People then think of looking for those numbers that make Mn a prime number. n. Such primes are called Mersenne primes. Elementary algebra knowledge tells us that unless n itself is a prime number, Mn will not be a prime number, so we only need to pay attention to taking the prime value of n. However, most prime numbers n also cause the Mersenne number Mn to be composite. It seems that finding the appropriate n is not easy--although the first few numbers may not seem difficult to you
. On February 12, 1998, 19-year-old Roland Clarkson of California State University found a new suitable n
. He used a computer to discover the largest known prime number. This prime number is 2 times 3021377th power minus 1. This
is a 909526-digit number. If this number is written down continuously in ordinary font size, its length can reach more than 3000
meters. Clarkson used his spare time to calculate for 46 days, and finally proved on January 27 that this was a prime number. How big is this
prime number? Let’s compare it with another large prime number!
In an ordinary 8×8 square chessboard, we place 2 mm thick chips on the square according to the following rules
(such as British 10 pence coins ). First number the squares from 1 to 64. Place 2 chips in the first grid, 4 chips in the second grid, and 8 chips in the third grid. By analogy, the number of chips placed in the next grid is exactly twice that of the previous grid. Therefore, there are 2n chips in the nth grid, and there are 264 chips in the last grid.
Can you imagine how high this stack of chips is? 1 meter? 100 meters? 10,000 meters? It’s definitely not right
! Well, believe it or not. This stack of chips will soar into the sky, surpassing the moon (which is only 400,000 kilometers away
), surpassing the sun (150 million kilometers away), and almost reaching the nearest star (other than the sun) Centauri. Alpha
Star, about 4 light years away from the earth. Expressed as a decimal number, 264 is: 18446744073709551616.
264 is so impressive. In order to get 23021377-1, which appears in the largest prime number at present, you need to
be on a chessboard larger than 1738×1738 squares. Play the game above!
Finding large prime numbers has practical application value.
It promotes the development of distributed computing technology. In this way, it is possible to use large numbers of personal computers to do projects that would otherwise require supercomputers. In addition,
In the process of finding large prime numbers, one must repeatedly multiply large integers. Now some researchers have found ways to speed up computing, and these methods can be used in other scientific research. Large prime numbers can also be used to encrypt and decrypt.
The method of finding Mersenne primes can also be used to test whether computer hardware operations are correct.
Compared to the infinite number of prime numbers, what we have discovered so far is only extremely limited. At the same time, we can
prove very few propositions about prime numbers. Goldbach's conjecture is a proposition about prime numbers, a proposition that we humans have not proven for more than 250 years.
Goldbach's conjecture
It seems to be a very simple number, but it contains a lot of interesting and profound knowledge. In number theory research, we often carefully propose "guesses" based on some perceptual knowledge, and then demonstrate them through strict mathematical deductions. We said above that any composite number can be decomposed into the product of prime numbers, so what about decomposing the composite number into the sum of prime numbers? Are there any rules here?
In 1742, Goldbach, a German middle school teacher, discovered that "any
large even number can be written as the sum of two prime numbers." For example: 6=3+3, 9=2+7 and so on. He verified many even numbers and they all proved to be correct. But this requires proof. Because mathematical propositions that have not yet been proven can only be called conjectures. He could not prove this proposition himself, so he asked Euler, a famous Swiss mathematician at the time, for help. Euler was one of the most famous mathematicians at the time. Although he believed in Goldbach's conjecture, he was stumped by this seemingly simple proposition. Until his death, Euler was unable to complete the proof of Goldbach's conjecture.
Goldbach's letter proposed two conjectures:
Any even number greater than 2 is the sum of two prime numbers.
Any odd number greater than 5 is the sum of 3 prime numbers.
It is easy to prove that conjecture (2) is a corollary of conjecture (1), so the problem boils down to proving conjecture (1).
In fact, for this conjecture, someone has verified the even numbers one by one. Until it reaches hundreds of millions, it shows that this conjecture is correct. But what about bigger and bigger numbers? The guess should be right. The conjecture
should be proven. However, it is very difficult to prove it. In 1900, the German mathematician Hilbert regarded Goldbach's conjecture as one of the most important remaining mathematical problems in the past in his speech at the International Mathematical Society. He included
"Goldbach's Conjecture" among his "23 Challenges for Contemporary Mathematicians". In 1912,
German mathematician Landau said in a speech to the International Mathematical Society that even if the weaker proposition "(3) exists a
positive integer a, such that every "An integer greater than 1 can be expressed as the sum of no more than a prime numbers", which is also beyond the reach of modern mathematicians. It should be noted that if (1) is true, just take a=3. In 1921,
British mathematician Hardy said at a mathematics conference held in Copenhagen that the difficulty of conjecture (1) is comparable to
any unsolved mathematical problem. .
However, human intelligence is always breaking through the limits set by themselves one after another.
Just one year later, in 1922, British mathematicians Hardy and Littlewood proposed a method to study Goldbach's conjecture, the so-called "garden method" ". In 1937, the Soviet mathematician I. Vinogradov applied the circle method and combined it with the trigonometric sum estimation method he created to prove that every sufficiently large odd number is the sum of three prime numbers
p>
.
This basically proves the conjecture (2) proposed in Goldbach's letter.
While some mathematicians are trying their best to attack Goldbach's conjecture (2), another group of mathematicians also sounded the clarion call for conjecture (1). A long time ago, people wanted to prove that every big even number is the sum of two "
not too many prime factors" integers. They want to set up the encirclement in this way, and want to gradually and step by step
prove the proposition of Goldbach's conjecture, that is, a prime number plus a prime number (1+1) is correct. So, step by step, although very slowly, people finally got closer to proving Goldbach's conjecture.
In 1920, the Norwegian mathematician Brown improved the more than 2,000-year-old Erardo-Nissl "sieve method" and proved that every sufficiently large even number is two The sum of positive integers whose number of prime factors does not exceed 9. Relative to the final
proposition (1+1), we record Brown's result as (9+9). In 1924, the German mathematician Rademacher
proved (7+7); in 1930, the Soviet mathematician Schniemann used the "density" combination of integers he created
The Brownian sieve method proves proposition (3) and can estimate the value of a. In 1932, the British mathematician Esterman
proved (6+6); in 1938, the Soviet mathematician Buchstab proved (5+5); in 194
In ○ year, he proved (4+4) again. In 1956, the mathematician Vinogradov proved (3+3)
.
Chinese mathematician Hua Luogeng began to study this problem as early as the 1930s and achieved very good results. He proved that for "almost all" even numbers, the conjecture (1 ) are all correct. Soon after liberation, he initiated and guided some of his students to study this issue, and achieved many results, which were highly praised at home and abroad. In 1965
Chinese mathematicians first showed their skills, and Wang Yuan proved (3+4). In the same year, Soviet mathematician A. Vinod
Gradov proved ( 3+3). In 1957, Wang Yuan proved (2+3). The encircling circle is getting smaller and smaller,
getting closer and closer to (1+1). But all the above proofs have a weakness, that is, none of the two numbers can be definitely prime.
In fact, mathematicians have already noticed this. So, they set up another encirclement,
that is, they tried to prove that "any large even number can be written as a prime number and another integer with not too many prime factors.
"In 1948, the Hungarian mathematician Lan Enyi reopened another battlefield and proved the shortcut: every big even number is a prime number and a "prime factor does not exceed six. "The sum of the numbers." In 1962, Chinese mathematician and Shandong University lecturer Pan Chengdong and Soviet mathematician Balba independently proved (1+5), making a step forward; in the same year, Wang Yuan , Pan Chengdong and Barbarn all proved (1+4)
. In 1965, Buchstab, Vinogradov and the mathematician Pompierre all proved (1+3)
.
The continuous progress people have made in proving Goldbach's conjecture seems to make people see the hope of completely proving it.
From (1+3) to (1+1), there are only two steps left. Who can finally take off this crown jewel?
In 1966, the young Chinese mathematician Chen Jingrun proved (1+2) and achieved the best result in the world so far regarding the conjecture
(1). He proved that any sufficiently large even number can be expressed as the sum of two numbers, one of which is a prime number and the other is either a prime number or the product of two prime numbers. Although "Goldbach's theorem" has not yet been produced, this conclusion closest to it has been unanimously named by the name of a Chinese person - "Chen's" by all countries in the world. theorem".
Pick the crown jewel
In 1933, Chen Jingrun was born in Fuzhou City, Fujian Province. His father was a clerk in the post office, and his mother was a kind but overworked woman. She gave birth to twelve children in one day and supported six. Although there is no set of parents who do not want to love their children, Chen Jingrun, who ranks third, has older brothers and sisters, and younger brothers and sisters, so he cannot be the most loved one by his parents. child. As if he was a superfluous person, Chen Jingrun did not enjoy much childhood joy.
When Xiao Jingrun just started to remember, the Japanese invaded Fujian Province. When he was young, he could only live in fear, and his soul was greatly hurt. He couldn't have fun at home, and he was always bullied in elementary school, which made him develop an introverted character. Chen Jingrun began to like mathematics, because the calculation of mathematical problems could help him kill most of his time.
After graduating from elementary school, Chen Jingrun was still a child who suffered discrimination in junior high school. After the Anti-Japanese War ended, Chen Jingrun entered Yinghua Academy. In the school at that time, there was a mathematics teacher who was once the dean of the Department of Aviation at National Tsinghua University. This teacher is knowledgeable and tireless in teaching, and inspired many students to love mathematics.
Once, the teacher introduced a famous problem in number theory to the students during class, which was Goldbach's conjecture.
For other students, maybe the three minutes of excitement passed quickly, because this is a problem that has troubled mankind for two centuries! Not to mention solving it, even for a great mathematician, it takes a lot of effort to make a little progress. However, I was fascinated by this problem and it was deeply imprinted in my mind, until I devoted my whole life to it!
After graduating from high school, Chen Jingrun entered the Department of Mathematics of Xiamen University. Due to his excellent grades, he graduated early, stood on the podium and became a teacher. However, his long-term introversion prevented him from imparting all his rich knowledge to his students like the teacher in high school. After many twists and turns, his mathematical talent was discovered by Hua Luogeng, who was working at the Institute of Mathematics of the Chinese Academy of Sciences at the time. Chen Jingrun was transferred to this temple of Chinese mathematical research in 1956. , became an assistant researcher.
Since then, his mathematical talent has been fully demonstrated. In just a few years, he improved the results of Chinese and foreign mathematicians in aspects such as the whole point problem in a circle, the whole point problem in a sphere and Waring's problem. Based on these achievements alone, he has achieved great success. But he has never forgotten the deep imprint left in his heart in high school - Goldbach's conjecture. After having sufficient conditions, he marched towards this pearl
!
Remitting efforts have yielded fruitful results. Chen Jingrun finally took another extremely important step on the road to picking up the pearl. After making important new improvements to the sieve method, he initially solved (1+2
) in 1965 and wrote a proof of more than 200 pages. In May 1966, Chen Jingrun announced in the seventeenth issue of "Science
Bulletin", a publication of the Chinese Academy of Sciences, that he had proved (1+2).
Just a year ago, foreign mathematicians proved (1+3) using high-speed computers. But Chen Jingrun came to a better conclusion just by
handwriting mental arithmetic. However, since the proof is too cumbersome, further simplification is needed.
So, Chen Jingrun plunged into the manuscript again and continued his climb. Nothing that has nothing to do with research can disturb his thoughts.
In his 6-square-meter hut, among several sacks of calculation paper, Chen Jingrun endured hardships that ordinary people could not endure, and tirelessly pursued that dream.
Just after the Spring Festival in 1973, Chen Jingrun completed a revised draft of his thesis "The table of large even numbers is the sum of the products of a prime number and
not more than two prime numbers", that is (1+2 ) and publish it. Chen Jingrun proved in the paper
:
Each big even number can be expressed as the sum of a prime number and the product of no more than two prime numbers;
Suppose D(N) is the number of representations in which N is the sum of two prime numbers. It is proved that for a sufficiently large even number N, D(N)<7.8342(
N)/(LnN)2; < /p>
These two conclusions have greatly advanced the proof of Goldbach's conjecture, and are known internationally as "Chen's
theorem."
This achievement aroused strong repercussions in the world of mathematics and won our country a huge international reputation. Western reporters quickly learned of the incident, and the news quickly spread around the world. When the British mathematician Hubblestein and the German mathematician List learned of this matter, the book "Sieve Method" was being printed. However, they immediately took back the manuscript and re-edited it
and added Chapter 11: "Chen's Theorem" and gave it a very high evaluation: "From any aspect of the sieve method
In other words, it is the pinnacle of glory." At the same time, in some foreign mathematics publications, there are countless similar words of praise such as "outstanding achievements
", "brilliant theorems" and so on. A British mathematician even wrote to him
"You moved mountains!"
It is regrettable that the long-term and painstaking research has brought negative effects to Chen Jingrun's body. suffered many illnesses. Although he
received the cordial care of the party and the country, he was still unable to take the step to prove Goldbach's conjecture due to mental and physical difficulties.
Mathematicians from all over the world continued to struggle for it. It was the last step in a classical mathematics problem that lasted for more than 250 years, leaving behind the biggest regret in the history of mathematics in this century. Despite this, Chen Jingrun has conquered six or seven of the more than 30 world-wide number theory problems on his own, especially his achievements in proving Goldbach's conjecture. >
Today, no one can match it.
March 19, 1996, is a regrettable day for the entire world of mathematics.
Professor Chen Jingrun, an academician of the Chinese Academy of Sciences and a first-level researcher at the Institute of Mathematics, passed away at the age of 63 due to long-term illness and ineffective treatment.
The expectations of the century
Many people don’t understand what is the significance of studying such a “pure numbers game” as Goldbach’s conjecture
? You know, scientific achievements can be roughly divided into two categories. One type has obvious economic value and can be directly calculated by the amount of material wealth. It is a "treasure of value"; however, the other type of achievement is in the macro world and micro world.
< p>The economic value obtained from the world, cosmic bodies, elementary particles and other fields is incalculable and far beyond people's imagination. They are called "priceless treasures". Chen Jingrun's "Chen's Theorem" belongs to the latter.Goldbach's conjecture is very important to mathematics. In fact, as one of the most important conjectures about prime numbers, the "basic particle" of mathematics, solving it will It will make the entire humankind's understanding of natural science a big step forward. Therefore, many mathematicians are committed to simplifying the proof of "Chen's theorem". At present, there are several simplified proofs in the world. The simplest one was obtained by Chinese mathematicians Ding Xiaqi, Pan Chengdong and Wang Yuandong.
Many methods invented and applied in the process of human research on Goldbach's conjecture not only have
wide application in number theory, but can also be used in many branches of mathematics. It has promoted the development of these branches of mathematics and provided endless impetus for the progress of the entire society. For example, prime numbers provide humans with a good way to compile passwords
and play a great role in the security of people's communications. Mathematics, as the cornerstone of the building of natural science, every progress, even if it is extremely small, may enable us to build the entire building more brilliant and spectacular.
Decades have passed. Although attempts to prove Goldbach's conjecture have never stopped since the day it was proposed,
the whole world has Once again, I fell into confusion for a long time. Now
mankind once again stands at the historical moment at the turn of the century. The rapid development of science and technology has provided scientists with far more convenient conditions for climbing
the peak of knowledge. Especially the use of high-speed computers has made it possible to solve some mathematical problems such as the "four-color theorem". But for the crown jewel of Goldbach's conjecture, will human ingenuity be able to fully reveal its dazzling halo in the next century?
No one knows the answer, and the hope of the century is calling to mankind. (Pan Zhi)