“Ah … no kidding.” The cloud of unease that had been spreading in Ishigami’s chest vanished in a moment. “Now that you mention it, I remember him looking at a letter that came to me from the university. That must’ve been why he asked about it. Wonder why he didn’t tell me he was a fellow alum?”
“Well, honestly, he doesn’t consider graduates from Imperial University sciences to be his classmates. Sometimes, I don’t even think he thinks of us as the same species.”
Ishigami nodded. He felt the same way about those in the humanities. It was strange to think of the detective as someone who had been at the same university at the same time.
“So Kusanagi tells me you’re teaching math at a high school?” Yukawa asked, staring directly at Ishigami’s face.
“The high school near here, yes. You’re at the university, Yukawa?”
“Yeah. Lab 13,” he replied simply.
Yukawa wasn’t trying to ring his own bell, Ishigami realized; he didn’t seem to have any desire to boast.
“Are you a professor?”
“No. I’m just futzing around as an assistant professor. It’s pretty crowded at the top, you know,” Yukawa said, without any discernible ire.
“Really? I figured you would be a full professor for sure by now, after all that hype about those magnetic gears of yours.”
Yukawa smiled and rubbed his face. “I think you’re the only one who remembers all that. They never did make a working prototype. The whole thing ended as an empty theory.” Yukawa picked up the sake bottle and began to open it.
Ishigami stood and brought two cups from the cupboard.
“But you,” Yukawa said, “I had you pegged as a university professor, holed up in your office, taking on the Riemann hypothesis or some such. So what happened to Ishigami the Buddha? Or are you truly following in the footsteps of Erdős, playing the itinerant mathematician?
“Nothing like that, I’m afraid,” Ishigami said with a light sigh.
“Well, let’s drink,” Yukawa offered, ending his questions and pouring Ishigami a glass.
The fact of it was, Ishigami had planned on devoting his life to mathematics. After he got his master’s, he had planned to stay at the university, just like Yukawa, earning his doctorate. Making his mark on the world.
That hadn’t happened, because he had to look after his parents. Both were getting on in years and were in ill health. There was no way he could have made ends meet for all of them with the kind of part-time job he could have held while attending classes. Instead, he had looked around for steadier employment.
Just after his graduation, one of his professors had told him that a newly established university was looking for a teaching assistant. It was within commuting distance of his home, and it would allow him to continue his research, so he’d decided to check it out. It was a decision that quickly turned his life upside down.
He found it impossible to carry on with his own work at the new school. Most of the professors there were consumed with vying for power and protecting their positions, and not one cared the least bit about nurturing young scholars or doing groundbreaking research. The research reports Ishigami slaved over ended up permanently lodged in a professor’s untended in-box. Worse still, the academic level of the students at the school was shockingly low. The time he spent teaching kids who couldn’t even grasp high school level mathematics had detracted enormously from his own research. On top of all this, the pay was depressingly low.
He had tried finding a job at another university, but it wasn’t easy. Universities that even had a mathematics department were few and far between. When they did have one, their budgets were meager, and they lacked the resources to hire assistants. Math research, unlike engineering, didn’t have major corporations waiting in line to sponsor it.
Ishigami had soon realized he had to make a change, and fast. He had decided to take his teaching credentials and make those his means of support. This had meant giving up on being a career mathematician.
He didn’t see any point in telling Yukawa all this, though. Most people who had been forced out of research had similar stories. Ishigami knew his was nothing special.
The sushi and sashimi arrived, so they ate, and drank a little more. When the bottle of sake Yukawa had opened was dry, Ishigami brought out some whiskey. He rarely drank much alcohol, but he did like to sip a little to ease his head after working on a particularly difficult mathematics problem.
Though the conversation wasn’t exactly lively, he did enjoy discussing their old school days, as well as a bit about mathematics. Ishigami realized how little of the last two decades he had spent just chatting. This might’ve been the first time he had talked this much to another person since graduating. Who else could understand him but Yukawa? Who would even recognize him as an equal?
“That’s right, I almost forgot the most important thing I wanted to show you,” Yukawa said suddenly, pulling a large brown envelope from his paper bag and placing it in front of Ishigami.
“What’s this?”
“Open it and find out,” Yukawa said with a grin.
The envelope held a sheet of paper covered with mathematical formulas. Ishigami glanced over it, recognizing it almost instantly. “You’re trying a counterexample to the Riemann hypothesis?”
“That was quick.”
The Riemann hypothesis was widely considered to be one of the most important unresolved problems confronting modern mathematics. The challenge was to prove a hypothesis proposed by the German mathematician Bernhard Riemann; no one had been able to do it so far.
The report Yukawa brought was an attempt to show that the hypothesis was false. Ishigami knew there were powerhouse scholars elsewhere in the world trying to do this very thing. Of course, none had succeeded yet.
“One of the professors in our math department let me copy this. It hasn’t been published anywhere yet. It’s not a complete counterexample, but I think it’s heading in the right direction,” Yukawa explained.
“So you think that the Riemann hypothesis is wrong?”
“I said it was heading in the right direction. If the hypothesis is right, then of course it means there’s a mistake in this paper.”
Yukawa’s eyes glittered like those of a young miscreant watching a particularly elaborate practical joke unfold. Ishigami realized what he was doing. This was a challenge. He wanted to see just how soft Ishigami the Buddha had grown.
“Mind if I take a look?”
“That’s why I brought it.”
Ishigami pored over the paper intently. After a short while he went to his desk and got out a fresh piece of paper. Laying it down before him, he picked up a ballpoint pen.
“You’re familiar with the P = NP problem, right?” Yukawa asked from behind him.
Ishigami looked around. “You’re referring to the question of whether or not it is as easy to determine the accuracy of another person’s results as it is to solve the problem yourself—or, failing that, how the difference in difficulty compares. It’s one of the questions the Clay Mathematics Institute has offered a prize to solve.”
“I figured you might be.” Yukawa smiled and tipped back his glass.
Ishigami turned back to the desk.
He had always thought of mathematics as a treasure hunt. First, one had to decide where to dig; then one had to determine the proper excavation route that led to the answer. Once you had a plan, you could make formulas to fit it, and they would give you clues. If you wound up empty-handed, you had to go back to the beginning and choose another route. Only by doing this over and over, patiently, yet boldly, could you hope to find the treasure—a solution no one else had ever found.
Therefore, it would seem that analyzing the validity of someone else’s solution was simply a matter of following the routes they had taken. In fact, however, it was never that simple. Sometimes, you could follow a mistaken route to a false treasure, and proving that it was false could be even harder than finding the real answer.