A genius? I just love studying.

Chapter 265 An Unexpected Joy, The Twin Prime Conjecture

Time seemed to stand still all of a sudden. Life in the small town was even more peaceful than Chen Hui had imagined. Every day, Chen Hui ran, recited poetry, read papers, and in his spare time, he attended seminars and discussed the latest achievements in cutting-edge mathematics with young geniuses from Princeton.

Deng Leyan has arrived on the Princeton campus and begun to complete the task assigned to him by Chen Hui.

Michael decided to come back after the semester started, saying he wanted to throw a big party to celebrate his acceptance to Princeton.

Chen Hui eventually recruited this eccentric but math-loving young Black man.

If those things hadn't happened, doing academic research in this small town might have been a good thing.

In his office, Deng Leyan was still reading papers. Although he had chosen mathematical analysis as his research direction, he had no idea what specific research topic he would pursue. These days, he would attend seminars, but the most important thing was to brush up on his basic knowledge.

Elena has begun research on applying the Navier-Stokes equations to weather forecasting, but she has encountered many difficulties and is visibly overwhelmed these days, even her pale blue eyes have become much dimmer.

Chen Hui's eyes were not bright either. He had been sitting at his desk for three days and three nights. The draft papers on the desk were piled up like a mountain, and many of them were even scattered on the ground.

Elena and her companion did not tidy up his draft paper; they did not want to interrupt their teacher, who was deep in thought.

The draft paper is interspersed with the imaginary part γn of the zero of the ζ function and the energy level formula of the quantum harmonic oscillator, En=ω(n+21)vsγn2πn/ln(n). These days of research on the Riemann Hypothesis have not been fruitless.

He came up with a crazy idea: if the non-trivial zeros of the zeta function are regarded as countless quantum harmonic oscillators on the complex plane, can their collective behavior reveal the absoluteness of the critical line?
Montgomery's correlation conjecture suggests that the distribution of zeros is similar to the eigenvalues ​​of a random Hermitian matrix. Previous studies, such as Hardy and Selberg, have been dedicated to proving that there are infinitely many zeros or proportional zeros on the critical line, but they have never been able to rule out the possibility of "a small number of zeros deviating".

By proving that the critical line is the only quantum ground state of the zero-point distribution, any deviation will cause the system's energy to diverge, leading to a probability tsunami. This directly eliminates the physical possibility of "exceptional zero points." Just as water always flows downhill, the zero point must be stable at the critical line—this is a mathematical law of complex plane spacetime!
This allows the Riemann Hypothesis to be transformed into a stability theorem for quantum systems, bypassing the technical bottlenecks of traditional analysis.

Chen Hui felt that he had reached the threshold of proving the Riemann Hypothesis. The rapid increase in his proficiency seemed to indicate that he was on the right track. In the past few days, he had been frantically pushing forward the verification.

But the road ahead is not smooth.

Looking at the deduction process on the draft paper on the table, Chen Hui fell into deep thought.

Suddenly, a set of unusual data caught his eye. When calculating the repulsive force between zeros, he discovered that when the imaginary part t→∞, the sum of the squares of the reciprocals of the distance δn between adjacent zeros, ∑1/δn^21, converges to 6/π2!

这让他猛然想起哈代-李特尔伍德猜想中孪生素数的分布常数孪生素数计数π2(x)2C2∫2x(lnt)2dt，其中C2=∏p>2(1(p1)21)=0.66016^
A related formula flashed into Chen Hui's mind like lightning.
6π2=ζ(2)=∏p(1p2)1，而C2=21∏p>2(p1)2p(p2)
分母中的(p-1)^2(p1)2与ζ(2)的欧拉乘积结构惊人相似！
The twin prime conjecture!

A flash of inspiration struck Chen Hui, and a piece of evidence immediately came to mind.

Chen Hui realized that the twin prime pair (p, p+2) can be encoded as an ideal class of the imaginary quadratic field Q(p(p+2). When both p and p+2 are prime numbers, the structure of this class exhibits a special symmetry.

With swift strokes of his pen, Chen Hui once again rapidly worked out his ideas on the draft paper.

He never expected to have such a pleasant surprise while studying the Riemann Hypothesis.

But this was to be expected. In recent days, he had studied a large number of documents on analytic number theory and had also done some research on the twin prime conjecture.

"Teacher, why don't you take a rest?"

Elena came to Chen Hui's desk and said with concern, "You haven't rested for three days. The Riemann Hypothesis can't be completed in a day. You need to ensure you are in good health so that you can persevere until the day you complete the proof."

Deng Leyan also stood beside Elena, silently expressing his support for her.

Chen Hui wondered if Deng Leyan was just naturally quiet, but he felt that Deng Leyan was even more aloof than when he participated in the competition a few years ago.

Chen Hui looked up, hesitated for a moment, then put down his pen and nodded in agreement, "Okay."

The framework for proving the twin prime conjecture has been established; the next step is simply to summarize and organize it, so there's no need to rush.

He suddenly hoped that Michael would arrive sooner, otherwise he could take over the work. Although he had two students, one was good at partial differential equations and the other wanted to study numerical analysis.

Elena hadn't expected Chen Hui to agree so easily. Recalling Chen Hui's bright smile, she suddenly had a crazy guess: "Could it be that the teacher has proven the Riemann Hypothesis?"

"That's not true."

Chen Hui shook his head, "It was just some unexpected gains."

Congratulations, teacher.

Elena was genuinely happy for Chen Hui. She knew that her teacher had only been studying the Riemann Hypothesis for less than a month, and proving it was obviously nonsense, but even a small breakthrough was something to be happy about.

Deng Leyan glanced silently at Chen Hui, then turned back to his seat and focused on reading the paper.

"Isn't this a bit too exaggerated? A small breakthrough already?"

"Could he have already proven the Riemann Hypothesis before I even graduate?"

"Of the seven millennium problems, Perelman proved one, and Chen Hui is about to prove the third. There are only three left. There are not many millennium problems left for me."

"Deng Leyan, you'd better work harder!"

No one knows what kind of activities Deng Leyan was carrying out beneath that poker face.

Chen Hui was in a good mood. He tidied up the draft a little, got up and walked towards the club. In the past three days, he hadn't eaten much because he was working on perfecting the theory of "critical line quantum measurement".

"Professor Chen, why haven't I seen you running these past few days?"

As soon as I stepped out of the office, a middle-aged man who looked like a rugby player greeted me warmly in the hallway.

This is Britt Adams from the Institute for Advanced Study, a professor of molecular biology. He's Chen Hui's running buddy at Carnegie Lake, though Adams runs 10 kilometers while Chen Hui runs 5 kilometers.

Chen Hui replied with a smile, "I've had some new inspiration recently, and I've been working on it in seclusion for a few days."

"Oh?"

"Is there any progress?"

Adams was very curious about this new professor in the Princeton mathematics department. Chen Hui himself probably didn't even know how famous he was in academia.

"There has been some minor progress."

"You've proven the Riemann Hypothesis?" Adams exclaimed in surprise.

Chen Hui smiled wryly, "Of course not, but there were some unexpected gains."

"What results?"

You should know in a few days.

Chen Hui doesn't have the habit of popping champagne halfway through a presentation. Now that he's already at Princeton, he plans to publish the proof of the twin prime conjecture directly in the Annals of Mathematics.

"Oh my God, you mathematicians have absolutely no sense of humor."

Adams waved his hand dismissively, looking disappointed, and walked towards Fefferman's office.

Chen Hui continued walking towards the club. He had originally planned to study biology and improve his biology proficiency, hoping to collaborate with Professor Adams later. Professor Adams was highly accomplished in molecular biology and had published two high-quality articles in *Nature*. However, with the Riemann Hypothesis making some progress, he didn't have time for anything else for the time being. He would put the biology proficiency matter aside for now and deal with it when he had more free time.

With his plans in mind, Chen Hui had already entered the club.

"Thank you, Professor Chen!"

As soon as he entered, a curly-haired boy came up to him and bowed sincerely to Chen Hui.

Chen Hui vaguely remembered that this was Ella, the boy he had met at the club that day. Presumably, the problem that had been troubling him had been solved.

"You're welcome. I didn't do anything; you just had the insight yourself."

Chen Hui waved his hand, picked up his plate, and began to devour the suspended dining area. He had only given a slight hint that the other party was able to complete the proof because he had already accumulated sufficient experience in this area. What he did was just add the last straw.

Even without him, Ella would eventually solve the problem, so he didn't dare take credit.

"If it weren't for you, I wouldn't have been able to solve this problem so quickly..." Ella continued to chatter on and on beside Chen Hui, but Chen Hui could understand her excitement and didn't stop her.

It turns out that after that day, in the following days, Ella built and improved a new framework based on "intrinsic oscillation phase management under dynamic renormalization" with astonishing efficiency, successfully overcoming the core obstacle of the paper.

His paper was eventually published in *Chaos, Solitons & Fractals*, a top journal in the subfield of ecological dynamics and nonlinear modeling.

The note, carefully framed and covered with coffee stains, syrup fractals, and the formula he had suddenly realized, was hung in the most conspicuous spot in his dormitory.

He deliberately showed the photos to Chen Hui to boast about, which was undoubtedly a badge of honor in his academic career.

As Chen Hui ate his casserole with gusto, he listened to Ella venting her excitement, while the table quickly filled up with people.

Chen Hui found it strange. It wasn't mealtime, so there shouldn't be so many people in the club. Moreover, these people weren't there to eat; instead, they were all holding notebooks and draft papers, placing them on the table, their eyes fixed on Chen Hui like hungry little birds, which was quite odd.

After eating and drinking his fill, Chen Hui quickly left the club with its strange atmosphere.

Ella finally stopped following him.

Standing at the club entrance, Ella looked somewhat dejected, but she didn't chase after him.

Chen Hui understood what the little guy meant. He had talked to Chen Hui for so long simply to win Chen Hui's favor and get him to take him on as a student.

However, Chen Hui's enrollment quota is already full, and he already needs to put in a lot of effort to recruit three students, so Chen Hui has no plans to recruit any more students for the time being.

"Hey Allen, wanna go to the club for dinner today?"

"Of course!"

"It's a pity I haven't seen Professor Chen for several days."

"Who says otherwise? Ella is such a lucky guy. His paper has already been published in a top journal. My teacher used to say that my research progress was faster than his."

"Jealousy makes me do Euler's integrals!"

"Don't be jealous. Go have some afternoon tea first. Maybe Professor Chen will show up today. Remember to bring your notebook and pen. Maybe you'll be the next one to receive the 'coffee stain revelation'."

"That makes sense, wait for me..."

Not long after leaving the club, Chen Hui overheard two classmates discussing something in the corridor. He immediately understood why so many people had been watching him while he was eating, and wondered how long they had been squatting there.

Chen Hui shook his head and chuckled, "A coffee stain as a clue?"

These people really have a knack for making up stories. It won't be long before this becomes one of the most talked-about campus tales on Princeton, just like Nash's chair and von Neumann's echo.

Indeed, the ultimate end of science is metaphysics!
Your Chinese language proficiency has improved from Level 2 (99%) to 100%, and you have now reached Level 3.

I don't know if it was because I suddenly heard this strange news, but a comment lit up: "After a lot of hard work, my Chinese language level has finally reached level 3."

[Congratulations, host! Your six subjects have been upgraded to level 3, and you have gained 1 free attribute point.]

"Ok?"

"Where did these six subjects at level 3 come from?"

He remembers that he was only at level 2 in subjects such as biology. Even though he improved his Chinese to level 3, he was only at level 3 in five subjects: mathematics, English, physics, and chemistry.

Chen Hui blankly brought up the data panel.
[Host: Chen Hui
Chinese Level 3 (0%)

Mathematics Level 5 (71%)

English Level 3 (90%)

Physics Level 4 (87%)

Chemistry Level 3 (26%)

Biology Level 2 (0%)

Physical Education Level 2 (13%)

Geography Level 2 (0%)

History Level 2 (0%)

Politics Level 4 (93%)

He quickly discovered the key point.

In the Politics category, it was clearly Level 4, and because of a series of recent events, his Politics level was almost at Level 5.

Unexpectedly, after reaching level 3 in his subject, he could gain free attribute points again as long as he passed six subjects. He also never expected that politics, which seemed useless, could be used to earn free attribute points.

What a pleasant surprise!
Biology might have to be prioritized a bit longer. Instead, you can focus on raising Chinese, English, and Chemistry to level 4, which will grant you another free attribute point.

With a powerful memory, coupled with insight, judgment, and creativity far exceeding the level of ordinary geniuses, improving in Chinese and English is not difficult, but chemistry still requires some effort.

"Boost my memory!"

Chen Hui used the free attribute points he had just obtained without hesitation.

Once his judgment reached level 7, he had a clear goal.

If you don't know what to add, it means your judgment is not good enough. Adding judgment will solve the problem. But now, he has obviously made a choice, which means his judgment is sufficient for the time being.

(End of this chapter)