1 On June 7th, 742, Goldbach wrote to Euler, a great mathematician at that time, and put forward the following conjecture: "Any integer not less than 4 can be expressed as the sum of two or more prime numbers" (it is different from the current expression because Goldbach thinks that1is also a prime number, see the illustration in the copy of the letter).

Today's expression is

Any even number greater than 2 can be expressed as the sum of two prime numbers. (a) (for example, 4 = 2+2)

Any odd number not less than 9 can be expressed as the sum of three odd prime numbers. (b) (Example: 9 = 3+3+3)

Any odd number greater than 5 (even number is acceptable) can be expressed as the sum of three prime numbers. (c) (Example: 7 = 2+2+3; 6 = 2 + 2 + 2)

Among them, conjecture A is the expression used by Euler in his reply, which is called double Goldbach conjecture or strong conjecture, and conjecture B and conjecture C are called triple Goldbach conjecture or weak conjecture. Through elementary algebraic transformation, we can know the sufficient condition that A is B and C, that is, if A is correct, B and C are also correct.

The first breakthrough of this conjecture came from Russian vinogradov. He unconditionally proved that conjecture B was correct by using circle method and exponential sum estimation. He proved that every odd number large enough can be expressed as the sum of three odd prime numbers. Here, a sufficiently large lower limit can be expressed as about 10 to the 400th power. So the proof of conjecture B comes down to verifying every odd number less than this number.

In 1966, Chen Jingrun proved "1+2", that is, "any large enough even number can be expressed as the sum of the products of a prime number and no more than two prime numbers".

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Trying to prove

Like many famous unsolved mathematical problems, Goldbach conjecture has many claimed proofs, but it is not accepted by the mathematical community.

Because Goldbach conjecture is easy to be understood by outsiders, it has always been a very common goal of pseudo-mathematicians. They try to prove it, or sometimes try to overthrow it, using only high school mathematics. Like the four-color theorem and Fermat's last theorem. The latter two questions are easy to describe, but their proofs are generally not complicated.

Problems such as Goldbach's conjecture cannot be ruled out by simple methods, but the first proof cannot be easily obtained with a lot of energy spent by professional mathematicians on such problems.

From 6 = 3+3,8 = 3+5, 10 = 5+5, ...,100 = 3+97 =1+89 =17+83. In the 20th century, with the development of computer technology, mathematicians found that Goldbach conjecture still holds true for larger numbers. However, natural numbers are infinite. Who knows if a counterexample of Goldbach's conjecture will suddenly appear on a sufficiently large even number? So people gradually changed the way of exploring problems.

1900, Hilbert, the greatest mathematician in the 20th century, listed Goldbach conjecture as one of 23 mathematical problems at the International Congress of Mathematicians. Since then, mathematicians in the 20th century have "joined hands" to attack the world's "Goldbach conjecture" fortress, and finally achieved brilliant results.

The main methods used by mathematicians in the 20th century to study Goldbach's conjecture are screening method, circle method, density method, triangle method and so on. The way to solve this conjecture, like "narrowing the encirclement", is gradually approaching the final result.

1920, the Norwegian mathematician Brown proved the theorem "9+9", thus delineating the "great encirclement" that attacked "Goldbach conjecture". What is this "9+9"? The translation of "9+9" into mathematical language means: "Any even number large enough can be expressed as the sum of two other numbers, and each of these two numbers is the product of nine odd prime numbers." Starting from this "9+9", mathematicians all over the world concentrated on "narrowing the encirclement", and of course the final goal was "1+ 1".

1924, the German mathematician Redmark proved the theorem "7+7". Soon, "6+6", "5+5", "4+4" and "3+3" were captured. 1957, China mathematician Wang Yuan proved "2+3". 1962, China mathematician Pan Chengdong proved "1+5", and cooperated with Wang Yuan to prove "1+4" in the same year. 1965, Soviet mathematicians proved "1+3".

1966, China mathematician Chen Jingrun conquered "1+2", that is, "any even number large enough can be expressed as the sum of two numbers, and one of these two numbers is an odd prime number and the other is the product of two odd prime numbers." This theorem is called "Chen Theorem" by the world mathematics circle.

Thanks to Chen Jingrun's contribution, mankind is only one step away from the final result of Goldbach's conjecture "1+ 1". But in order to achieve this last step, it may take a long exploration process. Many mathematicians believe that to prove "1+ 1", new mathematical methods must be created, and the previous methods are probably impossible.