Chapter 86: Understand?

Huh~

A bit tricky.

Su Kewei soon realized the initial proof method wouldn’t work; after a pause, he exhaled, pulled out another sheet of scratch paper, and prepared to construct an upper bound estimate for prime gaps.

Ten minutes passed.

Sweat began to bead on Su Kewei’s forehead; he had already piled up five or six sheets of discarded paper.

Xu Qingzhou saw that this senior would need some time, so he pulled out his phone and consulted Guo Zi on buying tickets to return home for National Day.

In the small group, Guo Zi was posting photos of himself and Ding Jiahui—both strolling around campus, with Guo Zi acting like a servant beside her.

Ding Jiahui: “Yao Yao, when are you coming to visit us? (Pitiful.)”

Song Yao: “Saturday.”

Ten minutes passed; the blackboard on the left was already filled with dense formulas. Gu Zhizhong turned to Xu Qingzhou and the others and asked, “Do you all understand this?”

Read! {

“Cramér’s Conjecture?” Xu Qingzhou promptly took the eraser and wiped clean the board’s recent writing.

In the Q&A room, several people had lifted their heads, glancing at the speaker with surprise—they’d just caught fragments of something about a conjecture.

“I ran into a tough problem and came over to ask.” Xu Qingzhou handed over the scratch paper directly.

“Understood.” Xu Qingzhou and Su Kewei nodded; Meng Bin thought for a moment, then nodded too.

Then construct an expression in terms of p such that when q is the smallest prime greater than p, we have qp ≤ f(p).

He said with some delight, “Senior, have you solved it?”

Gu Zhizhong gave a slight nod; after Xu Qingzhou finished erasing the board, he asked curiously, “How far did you get?”

Gu Zhizhong stood up, took the eraser, and said, “This problem is indeed difficult—or rather, it involves a famous mathematical conjecture: Cramér’s Conjecture.”

After about five minutes, Gu Zhizhong looked up at Xu Qingzhou and frowned, “Where did you encounter this problem?”

Gu Zhizhong took the paper, put on his glasses, and began studying it; beside him, Meng Bin silently picked up Gu Zhizhong’s teacup and headed toward the hot water room.

“Younger brother? Younger brother?”

“Yes, that’s one approach.” Gu Zhizhong nodded approvingly.

When Xu Qingzhou and the other two entered, Gu Zhizhong had just finished his task and looked surprised. “Oh, why did all three of you come together?”

Twenty minutes passed.

Su Kewei nodded and said, “I’ll take you there.”

“But there’s a problem: even if you find such a counterexample, it only proves the conjecture fails in that specific case—not that it fails universally.” Xu Qingzhou sighed, helpless.

Gu Zhizhong continued writing a long string of formulas, then said slowly, “Consider the average gap between consecutive primes; for large x, the average gap is roughly log x.”

“The Prime Number Theorem states that as x approaches infinity, the number of primes less than or equal to x, π(x), is approximately \frac{x}{\log x}.”

“Fine.” Xu Qingzhou sighed.

Xu Qingzhou: ?

Someone nearby greeted him softly. Xu Qingzhou replied, “I saw it in Sophie Dawson’s book, Advanced Mathematical Theory.”

In the group, everyone was planning activities for this Saturday.

The Q&A room was quiet, save for the soft scratching of chalk on the blackboard.

“This is a graduate-level textbook,” Gu Zhizhong smiled lightly, glancing at Su Kewei. “You’ve read it too, right?” Su Kewei nodded awkwardly—he hadn’t just read it, he’d even boasted he could solve it.

!.

Worried Xu Qingzhou and the others might be lost, he added, “This follows from the asymptotic expression of the Prime Number Theorem.”

Ding Jiahui: “Yao Yao, you just reminded me of a proverb—husband leads, wife follows.”

Meng Bin returned with the water.

Xu Qingzhou, who had been messaging in the group, heard someone calling him. He put down his phone and saw Su Kewei standing beside him.

Professor Gu used a novel approach: first introducing the Prime Number Theorem and the prime-counting function, then constructing a complex upper-bound function f(p) based on the properties of function z, which measures the “density” of primes less than p.

Xu Qingzhou pulled out a stack of scratch paper. “I tried several methods, but none worked. Eventually, I tried to find a counterexample to the conjecture—like a pair of consecutive primes (p, q) such that qp > 3×(log p)^1.5.”

As everyone knows, anything labeled a “conjecture” is never simple.

“Yes, this problem is still light-years away from the true difficulty of Cramér’s Conjecture—but it’s still no easy task.”

“I mostly understand this too,” Xu Qingzhou nodded.

Song Yao: “I’m in class. Talk to Xu Qingzhou about the plans.”

g(p)=3×(log p)^1.5+(p)π+1π≤g(π)(p, q)=(10007, 10009), gap = q-p = 10009-10007 = 2; 3×(log 10007)^1.5 ≈ 3×(9.2103)^1.5 ≈ 3×34.406 ≈ 103.22.

“Senior Su Kewei, Senior Meng Bin.”

This time, he simply wrote out the entire problem, hoping to spark collective brainstorming.

Gu Zhizhong explained his reasoning while swiftly writing on the blackboard with chalk.

Q&A room.

Ding Jiahui: “Yao Yao, you’re so cold (fragile).”

It was just a large lecture hall, with many seniors and juniors studying alone—mostly preparing for the postgraduate entrance exam. If they had questions, they could ask directly; every day, professors who didn’t agree with the curriculum would come here.

“Which one? Currently none,” Su Kewei blushed, noticing Xu Qingzhou’s skeptical look, and hurriedly added, “I study probability theory and mathematical statistics—yes, that’s it. I’m not strong in theoretical math. You should still consult Professor Gu.”

“I’m coming too!” Meng Bin put down his pen, eager to see how even Senior Su had been stumped—and how this problem could possibly be solved.

He felt a quiet admiration: each line of reasoning was its own domain. No wonder this professor, who had spent decades immersed in mathematics, had grasped the solution in minutes after struggling with it for two days.

Su Kewei also nodded; though the steps were obscure, he could still follow them. He looked at the younger brother beside him with new respect—this was only his first year, yet he was already tackling problems of this difficulty.

Xu Qingzhou and Su Kewei both nodded, but Meng Bin was clearly struggling—he’d already been confused during the upper-bound function explanation, let alone now.

At the same time, the seniors and juniors in the audience wore equally blank expressions.

Understand? Understand what?

(End of chapter)