Chapter Fifty-Six: Teacher?

Wu Tong never harbored the arrogant delusion that, as a mere beginner, she could challenge the world’s great mathematical enigmas. Yet, after thoroughly reinforcing her grasp on the fundamentals of every branch of mathematics before the New Year, and now continuing her studies in number theory at Jingda, she had naturally come across a wealth of materials and papers on various conjectures in the field. Of course, ordinary problems no longer troubled her, and it would be false to claim that she felt no interest in conquering the world’s mathematical conjectures—every student of mathematics harbors that ambition at heart.

Having recently read and studied numerous works and papers on number theory, she had inevitably begun a deep dive into the realm of prime numbers. It is rare for a devotee of number theory not to be captivated by primes, and she was no exception. Among integers greater than one, those divisible only by one and themselves are called prime numbers.

From fragmentary memories of her previous life, Wu Tong recalled that the latest major development on twin primes occurred around 2013, when Professor Zhang Yitang, teaching in America, proved that there exist infinitely many pairs of primes whose differences are less than seventy million. Although seventy million is an enormous number, this result marked the first time anyone had formally shown the existence of infinitely many pairs of primes separated by a fixed bound. Yet, as one marches forward on the road to the twin prime conjecture, progress becomes ever more arduous—Professor Zhang had advanced the field only to this point.

The International Mathematical Union, building on Professor Zhang’s methods, managed to reduce that bound from seventy million down to two hundred forty-six. Mathematicians across the globe, intrigued by the problem, continued to strive, hoping that with the right function, the gap could one day be narrowed to the elusive figure of two. However, the leap from two hundred forty-six to two still loomed like a towering mountain, stymieing mathematicians worldwide.

More regrettably, Professor Zhang’s techniques largely relied on classical mathematical tools and did not offer much in the way of genuine innovation, a point that sparked some controversy at the time. Because it was a breakthrough by a Chinese mathematician, the story dominated domestic headlines, and Wu Tong remembered it well.

If memory served, Professor Zhang had used a modified sieve method. Though Wu Tong had yet to find a path to solve the twin prime conjecture herself, her intuition—honed by her meditative stone of enlightenment—told her that the sieve method alone would not suffice. Professor Zhang’s contributions to the conjecture were indelible, yet even he could not unlock its ultimate solution.

After carefully working through the arguments presented in various papers using different methods, Wu Tong began to develop some ideas of her own. She jotted down a few lines of equations, following her own line of reasoning. Tentatively, she pressed on, attempting to push her thoughts a few steps further, but her ideas were still too scattered to form a coherent thread; there was much research and study left for her to do.

The twin prime conjecture was, as expected, a world-class enigma. Solving university-level problems posed little challenge to her now; she could extract the core of a question and follow its logic with ease. Yet, before this problem, she felt acutely the vastness of her own ignorance and the daunting, uncertain road ahead. Wu Tong had no intention of biting off more than she could chew—however difficult the question, there must be an answer. If she could not yet provide it, it simply meant she had more to learn.

Clutching her scribbled drafts, Wu Tong rose, intent on seeking more materials—sieve methods, the circle method, Lie groups, or perhaps something else. Her thoughts remained immersed in a continuous process of deduction, her steps guided entirely by instinct.

It was only the fifteenth day of the lunar new year, with a few days left before the semester began. The Jingda library was sparsely populated, especially in the mathematics section. Wu Tong always chose the same spot; by now, she could navigate the route with her eyes closed.

Still lost in contemplation, Wu Tong made her way by habit to the shelf she wanted, preparing to circle around for a particular reference. But as she turned the corner—having assumed this area would be deserted—she discovered someone directly opposite her, reaching for the same goal.

The library’s lighting cast a long shadow of the newcomer, which faintly overlapped with Wu Tong’s own. Both stopped just in time, sidestepping each other to avoid a collision. Yet, in the flurry, Wu Tong’s draft papers fluttered to the floor like falling snowflakes.

“Sorry, I wasn’t paying attention. I didn’t bump into you, did I?” She realized she had nearly collided with someone.

Snapped out of her daze, Wu Tong’s first instinct was to apologize. She shouldn’t have let her absorption in deduction consume her attention, trusting only her instincts.

Looking up, she found herself gazing at a face both cool and striking—so handsome it would draw every eye in the room, the kind of countenance that, in ancient times, would have inspired legends. Yet, although Wu Tong was not immune to the shock of beauty, her heart was largely unmoved—her devotion to mathematics had already claimed all her passion, save for her family.

After apologizing, she bent down to retrieve her scattered notes.

Lu Xiao pinched the bridge of his nose lightly; a pair of gold-rimmed glasses perched on his high nose lent him an air of refined scholarship and steady erudition. He, too, had let his guard down. Kneeling, he gathered the papers that had fallen near him, and in the process, a sequence of deduction formulas caught his eye. Primes?

“I was distracted as well. Is this the special case of the twin prime conjecture under the Bounded Gap Conjecture? …And you’ve hit a wall with the sieve method?” Glancing at the top sheet, he handed Wu Tong her drafts, his tone gentle and inquisitive.

His finely shaped brows moved slightly. How long had it been since he last left his research office? Were children nowadays already challenging the world’s greatest conjectures? Wu Tong’s youthful face left no doubt as to her age—was she the child of a professor in the mathematics department? A family of scholars, and a worthy successor.

“Yes, sir,” Wu Tong replied, unsure how to define the stranger’s identity, settling on the respectful title of “sir.” He didn’t have the air of a typical graduate student; if anything, he seemed too young to be a professor at Jingda, yet age was never the sole measure of ability. The ease with which he discerned her research direction from a glance at her scattered formulas bespoke formidable mathematical mastery—indeed, he seemed the most gifted mathematician she had encountered thus far.

Lu Xiao pondered for a moment, then adjusted his glasses.

“Perhaps the answer lies beyond the sieve method. Hilbert’s use of topological methods to prove the infinitude of primes—perhaps you could explore that direction.” He reached up to the top shelf and handed Wu Tong a book. “This is Zermelo’s supplementary research applying topology to sieve theory. I hope it might inspire you.”

Topology? The study of properties that remain unchanged when geometric shapes or spaces are continuously deformed? In that instant, the distances between the library’s lights, shelves, and the figure before her seemed to form a golden triangle in space; Wu Tong’s perspective filled with pure geometric images rotating in her mind, and the gentle glow of the enlightenment stele suffused her thoughts.

Ripples of inspiration stirred in her mind, as if a white dove of insight had taken flight. It seemed she had grasped a thread that could string together those scattered beads of thought into a beautiful necklace.

“Thank you, Professor,” Wu Tong said, accepting the book with gratitude.

She was now certain—such mastery could never belong to an ordinary student. With a few understated words, he had cleared away the fog that had clouded her vision, letting through a shaft of light. Even if she had guessed his title wrong, it did not matter; a true teacher is one who can guide another. To be worthy of that name, even for a moment, was honor enough.