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From 1729 to 1764, Goldbach kept correspondence with Euler for 35 years. In the letter 1742 to Euler on June 7th, Goldbach put forward the following conjecture: (a) Any even number ≥6 can be expressed as the sum of two odd prime numbers.

(b) Any odd number ≥9 can be expressed as the sum of three odd prime numbers. This is the so-called Goldbach conjecture.

He wrote in the letter: "My question is this:

Take any odd number, such as 77, which can be written as the sum of three prime numbers:

77=53+ 17+7;

Take an odd number, such as 46 1,

46 1=449+7+5,

It is also the sum of three prime numbers, and 46 1 can also be written as 257+ 199+5, or the sum of three prime numbers. In this way, I found that any odd number greater than 9 is the sum of three prime numbers.

But how can this be proved? Although the above results are obtained in every experiment, it is impossible to test all odd numbers. What is needed is general proof, not individual inspection. "

Euler wrote back: "This proposition seems to be correct". But he can't give strict proof.

At the same time, Euler proposed that this conjecture can have another equivalent version: any even number greater than 2 is the sum of two prime numbers, but he failed to prove this proposition. It is not difficult to see that Goldbach's proposition is the inference of Euler's proposition.

The original content of Goldbach conjecture can also be expressed as:

Any integer greater than 5 can be written as the sum of three prime numbers.

Today's common conjecture is expressed as Euler version, that is, any even number greater than 2 can be written as the sum of two prime numbers.

In fact, any odd number greater than 5 can be written in the following form: 2N+ 1=3+2(N- 1), where 2(N- 1)≥4. If Euler's proposition holds, even 2N can be written as the sum of two prime numbers and odd 2N+ 1 can be written as the sum of three prime numbers, then Goldbach's conjecture holds for odd numbers greater than 5.

But the establishment of Goldbach proposition does not guarantee the establishment of Euler proposition. So Euler's proposition is more demanding than Goldbach's proposition. Now these two propositions are collectively called Goldbach conjecture.

progress

Goldbach conjecture seems simple, but it is not easy to prove, which has become a famous problem in mathematics. In 18 and 19 centuries, all number theory experts did not make substantial progress in proving this conjecture until the 20th century. 1937 Soviet mathematician vinogradov (ииногралов, 189 1- 1983).

Consider expressing even numbers as the sum of two numbers, each of which is the product of several prime numbers. The proposition that "any big even number can be expressed as the sum of a number with no more than one prime factor and a number with no more than b prime factors" is recorded as "a+b", so the Coriolis conjecture is to prove "1+ 1" (that is, "any big even number can be expressed as a number with no more than 1 prime factor). 1966, Chen Jingrun proved that "1+2" was established, that is, "any big even number can be expressed as the sum of a prime number and another prime number whose factor is not more than 2".