Chapter 185: Challenging the Twin Prime Conjecture

Auditorium.

The audience was packed solid, with people even standing in the aisles; Xu Xu Qingzhou realized he had underestimated the appeal of the Cramér theorem.

He opened his prepared PPT and briefly outlined his presentation: "Today's talk is divided into three parts: statement of the theorem, proof of the theorem, and significance of the theorem."

"The problem of adjacent primes is a foundational issue in number theory; clarifying this property relates to the construction of all number systems. Euclid proved that as natural numbers extend, primes must extend as well."

Xu Xu Qingzhou focused on the proof, explaining how to use the Selberg sieve to precisely understand the distribution of primes, then how to refine and supplement it using analytic number theory.

"In the study of the Cramér theorem, the first step is to derive a functional expression for the gaps between consecutive primes."

"This is also the key to why the proof succeeds—the function f(p). You may find it unfamiliar; yes, I constructed it myself. It provides a measurable estimate of the 'density' of primes less than p."

"Through this function, I successfully widened the upper bound to c×(log p)^a. By the way, a is a real number greater than 1 but less than 2."

"Mr. Xu, on page five, third paragraph of your paper, you state 'squaring both sides yields (ln p_n)^2 ≈ (p_n/n)^2, setting the difference between p_n and its successor p_{n+1} as 2k'—this does not logically connect to what follows."

Read!

But it doesn't have to be this way.

Hmm, did it fail? Or is it unworkable? Doubts stirred among the audience; even Professor Maynard was curious—the proof of the twin prime conjecture is immensely complex, yet Xu Xu Qingzhou had only written one part.

He finished speaking and began packing up.

The last questioner was Professor Maynard.

Watching the young man on stage, Academician Xu felt his earlier worries were entirely unnecessary; this kid's presence on stage matched that of any preceding mathematician.

As he explained, he wrote the formulas on the blackboard.

On stage, Xu Xu Qingzhou put down his chalk and stopped writing. The source of inspiration had opened; the path ahead was clear, but the workload remained immense—impossible to finish here, he'd need to return to the hotel and think slowly.

All questions fell within Xu Xu Qingzhou's expectations, so he answered them swiftly.

Two minutes passed.

For this question, Xu Xu Qingzhou had prepared an answer; he smiled and said: "(p_n/n)^2 is divergent—that is, as n grows, the corresponding prime grows larger."

To put it vividly, before, his mind felt filled with disordered particles in motion—he could see them, but couldn't capture their traces.

After greeting Professor Maynard and Academician Xu, Xu Xu Qingzhou quickly left the auditorium, preparing to seclude himself at the hotel.

Professor Maynard stared at the screen, his expression grave, and said slowly: "Is he aiming for the twin prime conjecture?"

Another hand rose: "Mr. Xu, in your proof, you widened the upper bound to c×(log p)^a."

Someone asked eagerly: "Mr. Xu, are you attempting the twin prime conjecture?"

Meanwhile, some scholars exited the auditorium.

A young scholar hadn't caught on yet: "What is this?"

But then, suddenly, a "boson" condensed into a "Bose-Einstein condensate"—all mental distractions aligned instantly, pointing toward that brilliant spark of inspiration.

Keshan also stared intently at Qingzhou, growing more intrigued—perhaps because she had a tendency to admire strength? Elsewhere, the Indian young man scowled, his envy overwhelming; this Chinese man was terrifyingly strong. Still, he took a deep breath and lowered his eyes to the proof materials of the twin prime conjecture before him.

"It's the sieve method," a mathematician specializing in primes had already recognized it and answered.

Yes, he paid close attention to details—he always examined every detail when encountering a problem, which often revealed new insights, but this also meant he unconsciously overlooked the big picture, unable to see the whole like a "bird."

Leaving the noisy auditorium, Xu Xu Qingzhou slowed his pace.

Why couldn't he combine m and steps (10), (14), and (15) from his draft to "guess and infer" an asymptotic formula for the number of twin primes? Xu Xu Qingzhou squinted, picked up chalk amid the audience's surprised glances, turned, and began writing on the blackboard behind him.

Large mathematical talks often feature claims of proving some conjecture; people were used to such delusions. But this man was the prover of the Cramér theorem—that changed everything. Soon, the one giving the talk on stage would be Salman Khan! On stage, Xu Xu Qingzhou had entered his rhythm, recapturing the feeling of lecturing—explaining how to construct the mathematical model, then how he came to use statistical theory from physics, graph theory, and more to improve the entire proof.

Xu Xu Qingzhou's mind raced. Yes, he had once been blinded by details, fixated on finding an exact formula.

"Probably so," another mathematician nodded slightly.

Huh? Everyone froze, puzzled, turning toward the blackboard.

Xu Xu Qingzhou exhaled and looked at the silent crowd: "Ladies and gentlemen, the presentation is over."

Failed, definitely failed! The Indian young man screamed inwardly—the honor of proving the twin prime conjecture belonged to him! "Phew~"

For these questions, Xu Xu Qingzhou smiled faintly, leaving them with a mystery: "Truth will soon appear before us."

"Do you have an idea, Mr. Xu? I've studied the twin prime conjecture for over a decade—if you're willing, I've prepared all my research materials for you!"

Xu Xu Qingzhou's presentation was over—so what had he written? Everyone fell silent, staring at the young man on stage who had just written another string of formulas.

Compared to the excitement of others, Xu Xu Qingzhou stood frozen, as if struck by a bullet, motionless, lost in thought.

Professor Maynard began to clap; the hall erupted in thunderous applause.

"Xu, I just thought of Freudenthal's famous analogy."

【π(x)~l(s, \chi)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^s】

!.

Others could infer from the sieve method that it concerned the twin prime conjecture, yet it still looked like ancient script.

Sixty minutes passed quietly; the Q&A session had begun.

Zhao Zheng looked on with a silent sigh—having an overly brilliant junior brother was a heavy burden.

Professor Maynard's words were like a key unlocking his mind; the previously vague, hazy inspiration became crystal clear in an instant.

Professor Maynard had no questions about the Cramér theorem; instead, he said: "Mathematicians can be divided into birds and frogs. Birds survey the whole, contemplating macroscopic mathematical structures, while frogs dive into details, solving specific problems with strong practical skills."

As if the entire universe's symmetry and simplicity had opened its doors to you at that moment.

Someone else said: "Yes, it's the sieve method—a mathematical tool specifically used to study the distribution of primes."

At this moment, he felt no wild elation as he'd imagined; instead, his mind was calm. On his slow walk back to the hotel, he even let his thoughts go blank.

Thus, the audience largely understood Xu Xu Qingzhou's intention, their expressions filled with shock and doubt.

This habit of his was a double-edged sword—it helped him quickly locate key points, yet also made him forget to look ahead.

"In my view, you belong to the latter—you can use your sharp gaze to trace the marks of mathematical truth."

Five minutes later, the man said thank you and sat down, deep in thought.

"What! Attempting the twin prime conjecture?!"

"Hey, James—big news! Xu Xu Qingzhou from China has announced he will prove the twin prime conjecture!"

"Mr. Tucker, are you crazy?"

"No—he's the one who's crazy!"

"Mr. Tucker, calm down. I don't care who's crazy. But our front-page headline seems decided already!"

(End of chapter)