■ 1. Every even number not less than 6 is the sum of two odd prime numbers;

■2. Every odd number not less than 9 is the sum of three odd prime numbers.

In the letter 1742 to Euler on June 7th, Goldbach put forward a proposition.

He wrote: "My question is this: Take any odd number, such as 77, and you can write it as the sum of three prime numbers:

77=53+ 17+7;

Take an odd number, such as 46 1,

46 1=449+7+5 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 5 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 that this proposition seems to be correct, but he can't give a strict proof. At the same time, Euler put forward another proposition: any even number greater than 2 is the sum of two prime numbers. But he also failed to prove this proposition. It is not difficult to see that Goldbach's proposition is the inference of Euler's proposition. 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 number 2(N- 1) can be written as the sum of two prime numbers, and odd number 2N+ 1 can be written as the sum of three prime numbers, so Goldbach 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.