After the dinner, Pang Xuelin went directly to the hotel room where Zhang Yitang lived as agreed.

"Professor Pang, here, please come in!"

Zhang Yitang welcomed Pang Xuelin in.

Zhang Yitang lives in an executive suite with a special reception room.

The two came to the reception room, and Zhang Yitang went to make two cups of coffee first, and then walked over with a pile of manuscript paper.

"Professor Pang, here are some of my thoughts on Ponzi geometry theory in the past six months, you can help me see if there are any mistakes or omissions in them!"

"Good!"

Pang Xuelin took the manuscript paper and flipped through it.

There was silence in the reception room.

Time passed minute by minute, and it wasn't until half an hour later that Pang Xuelin raised his head and asked, "Professor Zhang, are you going to use the method of Ponzi geometry to prove the twin prime conjecture?" ”

Zhang Yitang nodded and said: "Under normal circumstances, a breakthrough in a major proposition will generally give birth to a new mathematical tool. However, when I proved the weakened version of the twin prime conjecture, I used a more traditional mathematical method, but after proving that the difference between the two twin primes was less than 70 million, I felt that the traditional method had reached its limit. Further down, I'm afraid you'll have to use some new math tools! ”

"I've been trying to build such a mathematical tool all these years, but now that I'm older, I don't have as much thinking and energy as I used to. It wasn't until the second half of last year, when your paper on Ponzi geometry came out, that I vaguely felt that Ponzi geometry was the key to solving the twin prime conjecture! ”

Pang Xuelin nodded.

Ponzi geometry explains how absolute Galois groups of rational numbers, or even flattened elementary groups of arbitrary algebraic clusters, affect the properties of the corresponding algebraic structures.

This theory essentially elucidated the nature of the structure of addition and multiplication, and bridged the gap between number theory and algebraic geometry.

It is of great significance for many conjectures in the field of number theory, such as Goldbach's conjecture, ABC conjecture, twin prime conjecture, hail conjecture, etc.

Zhang Yitang wanted to solve the twin prime conjecture problem through the relevant theory of Ponzi geometry, and Pang Xuelin was not surprised.

The so-called twin prime conjecture means that there are infinitely many prime numbers p, such that p+2 is prime. A pair of prime numbers (p,p+2) is called a twin prime.

This conjecture stems from the eighth of Hilbert's 23 questions, which was proposed by Hilbert at the International Congress of Mathematicians in 1900.

But more than 100 years later, this conjecture still haunts mathematicians around the world.

To date, the results of proving the twin prime conjecture can be broadly divided into two categories.

The first type is the so-called non-estimative results, the best results so far in this regard were obtained in 1966 by the late Chinese mathematician Chen Jingrun using the large sieve method.

Chen Jingrun proved that there are infinitely many prime numbers p, such that p+2 is either a prime number or a product of two prime numbers.

This result is very similar to the result of his conjecture about Goldbach.

At present, it is generally believed that due to the limitations of the sieve method itself, this result is difficult to be surpassed within the scope of the sieve method.

The second category is the estimative results, and the results achieved by Zhang Yitang belong to this category.

Such results estimate the minimum interval between adjacent primes, expressed mathematically, as: Δ:=limn→∞inf[(pn+1-pn)/ln(pn)].

Translated into the vernacular, this expression defines the interval between two adjacent primes to the logarithm of the smaller of them to the minimum value taken in the entire set of primes.

Obviously, if the twin prime conjecture is true, then Δ must be equal to 0.

Because the twin prime conjecture shows that pn+1-pn=2 holds for infinitely many n, and ln(pn) →∞, the minimum of the ratio of the two tends to zero for the twin prime set (and thus for the entire prime set).

However, it should be noted that Δ=0 is only a necessary condition for the twin prime conjecture to be true, but not a sufficient condition.

In other words, if Δ≠0 can be proved, the twin prime conjecture is not true; However, proving that Δ=0 does not mean that the twin prime conjecture is necessarily true.

Further international estimates of Δ began with Hardy and Littlewood.

In 1926, they used the circle method to prove that if the generalized Riemann hypothesis is true, then Δ≤2/3.

This result was later improved by Rankin to Δ≤3/5.

But both of these results depend on the generalized Riemann conjecture, which has not yet been proven, and can therefore only be regarded as conditional results.

In 1940, Bauer Eddis first gave an unconditional result using the sieve method: Δ<1.

Later, in 1955, in 1955, Bobby and Devinberg in 1966, and Huxley in 1977, this result was advanced to Δ≤15/16, Δ≤(2+√3)/8≈0.4665 and Δ≤0.4425, respectively.

Before Zhang Yitang, the best result of this method was the Δ≤0.2486 achieved by Meyer in 1986.

Zhang Yitang, on the other hand, has greatly advanced this result.

But even so, Zhang Yitang's previous work is still far from finally proving the twin prime conjecture.

"Professor Pang, do you think there are any problems or flaws in the thinking of these things written in my manuscript?"

Zhang Yitang asked.

Pang Xuelin shook his head and said: "It's hard to say, I don't have much research on the twin prime conjecture, I think you can try this idea, but I can't guarantee whether it will succeed." ”

Zhang Yitang smiled: "As long as there is no problem in the overall thinking." ”

Zhang Yitang hesitated for a moment and said, "Professor Pang, I have a reluctant request, I don't know if you are willing to agree?" ”

Pang Xuelin frowned slightly, nodded and said, "You say!" ”

Zhang Yitang said: "Professor Pang, I am sixty-six years old this year, to be honest, at this age, the main work has been put on helping to train the younger generation of mathematicians, and it is very difficult to go up the academic road." But I'm not reconciled, I've wasted too much time in my life, and I still hope to go up the academic path. So I want to ask you a favor, in the next two years, can you focus on tackling the twin prime conjecture, old man, I want to have an academic competition with you. If you can join, I'll be more motivated! When the time comes, whether it is you or I have proved this conjecture, it will be forgotten but my heart's greatest wish! If, after two years, we don't produce results, then the agreement will be automatically canceled! ”

Pang Xuelin laughed and said, "Professor Zhang, I promise you!" ”

Whether it is out of respect for this late-blooming and ambitious mathematician, or out of interest in this conjecture that has plagued mankind for 120 years, Pang Xuelin will not refuse.

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