Chapter Eighty-Two: Completion

In this regard, Wu Tong felt quite satisfied and confident. Her determination of the absolute direction reassured her that the inspiration she had received this time was indeed correct in its general course. The framework for verifying this general direction was already in place; now, it was just a matter of following the path step by step, filling in the details, completing the proof, and editing it into a paper. She could finish it within half a month—there were still twenty days until her birthday, so time was on her side.

Li and Cheng, on the other hand, were left utterly speechless, half a breath caught in their throats, nearly choking with disbelief. What… what did they just hear? Surely, they hadn't misheard. She already had the general direction of the proof? She planned to finish it before her birthday?

When was Wu Tong's birthday again?

July 29th!

Both had seen Wu Tong's information before, having helped her book tickets, and had seen her ID card. There could be no mistake about this detail. To complete the proof of the Twin Prime Conjecture within twenty days—were they mad, or was the world itself mad?

The breath they'd been holding finally was drawn back in with a gasp. Even when the other team members returned, they couldn't compose themselves. They had no time to ask how the others had performed, and in a daze, ushered everyone to dinner and then back to their hotel rooms.

"Keep all these manuscripts safe. The teachers are waiting for your good news!" Li Yisheng, having finally composed himself, reminded her carefully. At this point, nothing mattered more than Wu Tong completing the proof.

If Wu Tong truly succeeded in proving the Twin Prime Conjecture, it would be a historic sensation for Chinese mathematics, and likewise for the international mathematical community.

Compared to this, aside from the final IMO award ceremony—since they were already here, and had completed ninety-nine steps, it was only logical not to waste the last one—everything else could be set aside to make way for Wu Tong's proof.

"It's all right, Teacher Li. Relax, it's not something so important that I can't be disturbed. Let's stick to the original plan," Wu Tong replied with a laugh. This time, it wasn't as urgent as when she'd been struck by sudden inspiration and rushed to develop Zhou's proof in a single breath.

The Twin Prime Conjecture had been on her mind since the start of the year. For half a year now, she'd been focusing on it, and finally, inspiration had struck. With the direction of the final proof now certain, all that was left was to proceed methodically, fill in the gaps, and establish the proof.

"The sooner, the better!" After all, this was the Twin Prime Conjecture—a major world problem that had confounded mathematicians since 1900, for one hundred and ten years. In fact, ever since Euclid proved two thousand years ago that there are infinitely many primes, this problem had been gestating.

The sooner the proof was made public, and the name of a Chinese mathematician attached to it, the better.

What if someone else published it before Wu Tong? That would be a tremendous loss! Wu Tong had to publish her proof as soon as possible. If she wanted to travel around Kazakhstan afterwards, they could always charter a plane and assign someone to accompany her.

"All right, Teacher Li, I understand!" She would give it her all. Let the world witness the speed of China—she would aim for that goal!

"I'll upload the paper to arXiv as soon as possible." That was the fastest way to announce it to the world!

"If you need anything, just tell us. We'll be here these next few days!" Li Yisheng promised. He couldn't assist with the proof itself, but he would make sure everything else was taken care of.

Tomorrow, Professor Cheng would take the other students out for some fun, partly to distract the outsiders watching the Chinese team, and prevent any potential sabotage.

Even if he wasn't overly tense, Li Yisheng now viewed every outsider as a potential troublemaker.

After a simple meal, Wu Tong sat at her desk, took a deep breath, and picked up her pen. It was time to challenge herself to complete the remaining proof as quickly as possible.

This time, her thoughts did not stall out on the direction; her inspiration had constructed a bridge to the final proof. Strings of mathematical symbols no longer drifted in disorder, but, as if guided by some mysterious force, assembled into neat lines of equations under Wu Tong's pen.

She deduced, filled in, calculated—Wu Tong once again lost herself in the process. An endless storm of mathematical symbols whirled through her mind, while the luminous stele of enlightenment anchored her, fending off the chaos. In the deepest recesses, there was only the secret garden where all her knowledge and power converged.

Night deepened unnoticed; dawn's first light crept through the window lattice.

For every natural number K, there are infinitely many pairs of primes (p, p + 2k), K = 1... Proof established.

Once again, facing the morning light, Wu Tong let out a long breath. At last, the proof was complete.

She rubbed her brow. Two days and nights must have passed. On the night after the exam, she'd managed to get a little sleep, but last night, at the crucial stage, she'd worked through the night to finish the final deductions. Thankfully, she was full of energy, or she might not have held out.

She twisted open a bottle of water to moisten her dry throat, then, bathed in the morning light by the window, began her daily meditation and exercise. When she finished, the rosy glow of dawn had thoroughly replenished her vitality, and she felt radiant and refreshed.

After washing up, Wu Tong prepared to go out for breakfast. As Teacher Li had said, the milk and meat in Kazakhstan were excellent. Their hotel was of good quality, and there was a variety of options beyond just local cuisine—though most were Western. But after a night of hunger, and in a foreign land, Wu Tong was not about to be picky. Anything to fill her stomach would do.

These past days, Teacher Li had been bringing her meals to her room, but now that she had made real progress, she decided to get some fresh air before organizing her proof into a paper.

Wu Tong had just placed her hand on the doorknob when she realized that the door was being pushed open from the outside. She stepped back to give space for the door to open.

"Teacher Li? Professor Cheng, good morning!" Wu Tong greeted them with a smile.

"Oh, Wu Tong, you...?" Surprise welled up in Li Yisheng's chest. Since last night, Wu Tong had entered that unprecedented state again. He hadn't dared disturb her, but had been full of anticipation. Now, he had come early to check on her progress.

"Yes, just as you hoped, Teacher Li, Professor Cheng. I've finished the proof!" Wu Tong spread her hands lightly, her mood cheerful after conquering yet another great challenge.

"Fi...nished?" Joy burst from their chests—Li and Cheng nearly cried out, but managed to restrain themselves. Still, their happiness overflowed, impossible to suppress.

Wu Tong nodded in confirmation. "It's done!" She walked back into the room, handed the neatly arranged drafts to the two teachers who had followed her in. "Would you both help me verify it?"

"I'll organize the paper after I get something to eat. Right now, it's just the deduction process, a bit messy—please bear with it!"

"Wu Tong, have some food first. I'll review it with Professor Cheng," Li Yisheng said, placing the breakfast he had brought on the corner of the table. He took a tissue to wipe his hands, then accepted the stack of drafts from Wu Tong. To them, it felt as heavy as a mountain.

Heavens, just two days, was it? Since when had the proof of the Twin Prime Conjecture become something that could be finished in two days?

(End of chapter)