So, what is Goldbach conjecture?

Goldbach conjecture can be roughly divided into two kinds of conjecture:

■ 1. Every even number not less than 6 can be expressed as the sum of two odd prime numbers;

■2. Every odd number not less than 9 can be expressed as the sum of three odd prime numbers.

■ Goldbach correlation [

Goldbach C.,1690.3.18 ~1764.5438+01.20) is a German mathematician. Born in Konigsberg (now Kalinin); Studied at Oxford University in England; I originally studied law, and I became interested in mathematical research because I met the Bernoulli family when I visited European countries. I used to be a middle school teacher. /kloc-arrived in Russia in 0/725, and was elected as an academician of Petersburg Academy of Sciences in the same year. 1725 to 1740 as conference secretary of the Academy of Sciences in Petersburg; From 65438 to 0742, he moved to Moscow and worked in the Russian Foreign Ministry.

[Edit this paragraph] The source of Goldbach's conjecture

From 1729 to 1764, Goldbach kept correspondence with Euler for 35 years.

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, 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 these three prime numbers. 46 1 can also be written as 257+ 199+5, which is still the sum of three prime numbers. In this way, I found that any odd number greater than 7 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 put forward another proposition: any even number greater than 6 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. 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.

[Edit this paragraph] A brief history of Goldbach's conjecture

1742, Goldbach found in his teaching that every even number not less than 6 is the sum of two prime numbers (numbers that can only be divisible by 1 and itself). For example, 6 = 3+3, 12 = 5+7 and so on. 1742 On June 7th, Goldbach wrote to the great mathematician Euler at that time. In his reply on June 30th, Euler said that he thought this conjecture was correct, but he could not prove it. Describing such a simple problem, even a top mathematician like Euler can't prove it. This conjecture has attracted the attention of many mathematicians. Since Goldbach put forward this conjecture, many mathematicians have been trying to conquer it, but they have not succeeded. Of course, some people have done some specific verification work, such as: 6 = 3+3, 8 = 3+5, 10 = 5+5 = 3+7, 12 = 5+7,14 = 7+7 = 3+/kloc. Someone checked the even numbers within 33× 108 and above 6 one by one, and Goldbach conjecture (a) was established. But strict mathematical proof requires the efforts of mathematicians.

Since then, this famous mathematical problem has attracted the attention of thousands of mathematicians all over the world. 200 years have passed and no one has proved it. Goldbach conjecture has therefore become an unattainable "pearl" in the crown of mathematics. People's enthusiasm for Goldbach conjecture lasted for more than 200 years. Many mathematicians in the world try their best, but they still can't figure it out. Goldbach conjecture legend is actually the most legendary history in the history of science (see Baidu Goldbach conjecture legend for details).

It was not until the 1920s that people began to approach it. 1920, the Norwegian mathematician Brown proved by an ancient screening method, and reached a conclusion: every even n greater than it (not less than 6) can be expressed as the product of nine prime numbers plus the product of nine prime numbers, which is called 9+9 for short. It should be noted that this 9 is not an exact 9, but refers to any one that may appear in 1, 2, 3, 4, 5, 6, 7, 8 and 9. Also known as "almost prime number", it means that there are many pixels. There is no substantial connection with Goldbach's conjecture. This method of narrowing the encirclement is very effective, so scientists gradually reduce the prime factor in each number from (99) until each number is a prime number, thus proving Goldbach's conjecture.

At present, the best result is proved by China mathematician Chen Jingrun in 1966, which is called Chen Theorem: "Any large enough even number is the sum of a prime number and a natural number, while the latter is only the product of two prime numbers." This result is often called a big even number and can be expressed as "1+2". Professor Chen Jingrun's "sufficiently large" refers to the power of 10 of about 500,000, that is, adding 500,000 zeros after 1 is a number that cannot be tested at present. Therefore, Paul Hoeffmann wrote on page 35 of Revenge of Archimedes that a number that is almost prime enough is a vague concept.

■ Goldbach conjecture proves the relevance of progress

Before Chen Jingrun, the progress of even numbers can be expressed as the sum of the products of S prime numbers and T prime numbers (referred to as the "s+t" problem) as follows:

1920, Norway Brown proved "9+9".

1924, Latmach of Germany proved "7+7".

1932, Esterman of England proved "6+6".

1937, Lacey in Italy successively proved "5+7", "4+9", "3+ 15" and "2+366".

1938, Bukit Tiber of the Soviet Union proved "5+5".

1940, Bukit Tiber of the Soviet Union proved "4+4".

1948, Rini of Hungary proved "1+ c", where c is a large natural number.

1956, Wang Yuan of China proved "3+4".

1957, China and Wang Yuan successively proved "3+3" and "2+3".

1962, Pan Chengdong of China and Barba of the Soviet Union proved "1+5", and Wang Yuan of China proved "1+4".

1965, Buchwitz Taber and vinogradov Jr. of the Soviet Union and Pemberley of Italy proved "1+3".

1966, China Chen Jingrun proved "1+2".

All the above mathematicians won prizes in their own countries, but none of them were recognized by the International Mathematical Federation, so people began to think. Academician Wang Yuan made it clear in his speech at Nankai University in September 1986 that [1+ 1] and [1+2] are not the same thing. (See "Appreciation of World Famous Mathematical Problems" and "Hilbert's Tenth Problem" on page 188. Liaoning Education Press 1987 Edition). 1July 997 17, Academician Wang Yuan also said in CCTV's "Children of the East" program that Goldbach's conjecture only refers to 1+ 1. Academician Qiu Chengtong believes that literature, no matter how wonderful, cannot replace science. In 2006, Academician Qiu said that Chen Jingrun's success was caused by the media. It is generally believed that no one has made a substantial contribution to Goldbach's conjecture at present. All the proofs are problematic and have no substantive connection with Goldbach's conjecture.

It is found that (1+2) is much more difficult than (1+ 1) if almost prime numbers are removed. (1+3) is much more difficult than (1+2).

(1+ 1) is an even number (i.e. n >；) greater than the power of 1 of the first prime number "2" plus 1; 2+ 1) is the sum of a prime number and a prime number.

(1+2) is an even number greater than the second prime number "3", and 1 (that is, n > 3x3+ 1 = 10) is the sum of the products of one prime number and two prime numbers. For example, 12=3×3+3.

(1+3) is an even number (that is, n > 5x5x5+ 1 = 126), which is greater than the cubic addition of the third prime number "5" 1. For example, 128 = 5x5+3 = 5x5x3+53. Even numbers less than 128 cannot be expressed as (1+3), such as 4, 6, 8, 10, 12, 14,16,65438+.

(1+4) is an even number greater than the fourth prime number "7" plus 1 (that is, n > 7x7x7x7+ 1). For example, 2404=240 1+3. There are hundreds of even numbers less than 2404 that cannot be represented (1+4).

This is because the smaller the number of natural numbers, the more prime numbers, the less the composite number. For example, within 100, there are 25 prime numbers, 19 odd numbers with two prime factors, 5 composite numbers with three prime factors (27, 45, 63, 75, 99), and only 1 composite numbers with four prime factors (8/kloc-0 In fact, Goldbach conjecture is only the most difficult problem of this kind. Many problems are waiting for people to overcome.

It is impossible to prove "1+3" before "1+ 1".

Recognized by many scientists, in 1923, G.H.Hardy and J.E.Littlewood put forward the theory of R.

Asymptotic formula of (n): r (n) ∏ 2 [(p-1)] ∏ {1-[1((p-1) 2).

(lnn) 2} where: r(N) is the representation of an even number as the sum of two prime numbers, that is, n=p+p'

The number of prime numbers in even numbers that conform to Goldbach conjecture. ∏ represents the product of each parameter, ln table.

Stands for natural logarithm and 2 stands for square number. The parameter p of the first ∏ is greater than 2 and belongs to this one.

Prime numbers of even prime factors. The parameter p of second ∏ is a prime number greater than 2 and not greater than √ n.

The first value of ∏ is that the numerator is greater than the denominator and greater than 1. The value of second ∏ is a twin prime number.

Constant whose multiple of 2 is = 1.320 ... greater than 1. N/(lnN) is the number of prime numbers contained in the calculated n number.

(1/lnN) The ratio of prime number to number.

It is recommended that the asymptotic formula is greater than 1.

This paper discusses that the product of (the number of prime numbers contained in n numbers) and (the ratio of prime numbers to numbers) is greater than 1.

. A new formula for the number of prime numbers is derived: π (n) ≈ (0.5) (n0.5) [n0.5]/ln (n0.5)],

Get: N/(lnN)=(0.5) (square root number) (square root number)/(natural logarithm of square root number).

It is concluded that the number of prime numbers in n numbers is approximately equal to (half of the number of prime numbers in the square root number of n) and (of n).

Square root sign). N/(lnN) is the number of prime numbers contained in n numbers, and (1/lnN) is a prime number.

The ratio of number to number, the number of prime numbers is approximately equal to (the number of prime numbers half the square root) and (√N).

Product, the ratio of the number of prime numbers to the number is equal to {(the number of prime numbers half the square root) (√N)}/N,

It is approximately equal to (the number of prime numbers half the square root) divided by (√N).

{N/(lnN)}( 1/lnN) is approximately equal to the product of (half the number of prime numbers in the square root) and (√N), multiplied by (half the number of prime numbers in the square root) and divided by (√N). It is approximately equal to the square number (the prime number of half the square root).

As long as {the number of prime numbers half the square root} is greater than 1, n {(lnn) squared number} is greater than 1.

R(N)= (number greater than 1) (number greater than 1) = number greater than 1,

It can be clearly said that the number r(N) of even n expressed as the sum of two prime numbers is greater than 1.