The Science Fiction World of Xueba

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

"Professor Pang, here we come, please come in!"

Zhang Yitang welcomed Pang Xuelin in.

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

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

"Professor Pang, these are some of my thoughts on Ponzi's geometry theory in the past six months. Please help me to see if there are any mistakes or omissions in it!"

"good!"

Pang Xuelin took the manuscript paper and flipped through it.

The reception room fell into silence.

Time passed by, and it was not until half an hour later that Pang Xuelin raised his head and asked, "Professor Zhang, are you going to use Ponzi geometry to prove the twin prime number conjecture?"

Zhang Yitang nodded and said, "Under normal circumstances, a breakthrough in a major proposition will generally lead to the birth of a new mathematical tool. But when I proved the weakened version of the twin prime number conjecture, I used a more traditional mathematical method, but to prove that two twin primes After the difference between prime numbers is less than 70 million, I feel that the traditional method has reached the limit. Going down, I am afraid that some new mathematical tools must be used!"

"Over the years, I have been trying to build such a mathematical tool, but now that I am older, my thinking and energy are not as good as before. It was not until the second half of last year, when your paper on Ponzi geometry came out, that I I faintly feel that Ponzi geometry is the key to solving the twin prime number conjecture!"

Pang Xuelin nodded.

Ponzi geometry explains how the absolute Galois group of the rational numbers, and even the flattened fundamental group of any algebraic variety, affect the properties of the corresponding algebraic structures.

This theory essentially clarified the properties of additive and multiplicative structures, and built a bridge between number theory and algebraic geometry.

It is of great significance to many conjectures involving the field of number theory, such as Goldbach's conjecture, abc conjecture, twin prime conjecture, hailstone conjecture, etc.

Pang Xuelin was not surprised that Zhang Yitang wanted to solve the twin prime number conjecture problem through the relevant theories of Ponzi geometry.

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

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

But more than a hundred years later, this conjecture still puzzles mathematicians all over the world.

So far, the achievements in proving the twin prime number conjecture can be roughly divided into two categories.

The first category is the so-called non-estimative results. The best result so far in this area was 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, so that p+2 is either a prime number or the product of two prime numbers.

This result is very similar to his result on Goldbach's conjecture.

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 estimated results, and the results achieved by Zhang Yitang belong to this category.

This type of result estimates the minimum interval between adjacent prime numbers, expressed in mathematical language, is Δ:=l→∞f[(pn+1-pn)/ln(pn)].

Translated into vernacular, this expression defines the minimum value of the ratio between the interval between two adjacent prime numbers and the logarithmic value of the smaller prime number in the entire set of prime numbers.

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

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

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

In other words, if it can be proved that Δ≠0, the twin prime number conjecture is not valid; but the proof of Δ=0 does not mean that the twin prime number conjecture must be true.

Further international estimates of Δ began with Hardy and Littlewood.

In 1926, they used the method to prove that if the generalized Riemann conjecture 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, so they can only be regarded as conditional results.

In 1940, Paul Edith first gave a result without conditions Δ\u0026lt;1 by using the sieve method.

After that, Rich in 1955, Bobby and Devonburg in 1966, and Huxley in 1977, respectively advanced this result to Δ≤15/16, Δ≤(2+√ 3)/8≈04665 and Δ≤04425.

Before Zhang Yitang, the best result of this method was Δ≤02486 obtained by Mayer in 1986.

And Zhang Yitang took this result a step further.

But even so, Zhang Yitang's previous work is still far from the final proof of the twin prime conjecture.

"Professor Pang, do you think there are any problems or flaws in the thinking of the 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 number conjecture. I think you can try the idea you proposed, but I can't guarantee whether it will be successful or not."

Zhang Yitang said with a smile, "As long as there is no problem with the overall thinking."

Zhang Yitang hesitated for a moment and said, "Professor Pang, I have an unfeeling request. I wonder if you would agree to it?"

Pang Xuelin was slightly taken aback, nodded and said, "Say it!"

Zhang Yitang said, "Professor Pang, I am already sixty-six this year. To be honest, at this age, my main work has been to help train the younger generation of mathematicians. I want to go further on the academic road. , It is very difficult. But I am not reconciled, I have wasted too much time in my life, and I still hope to go up the academic road. So I would like to ask you to do me a favor, in the next two years, can you Focus on overcoming the twin prime number conjecture, old man, I want to have an academic competition with you. If you can join in, I will be more motivated! Whether it is you or me who proves this conjecture, Forget about the biggest wish in my heart! If after two years, we don’t produce any results, then this agreement will be canceled automatically!”

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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