“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.
Which was why someone had proposed the exasperating P = NP problem.
Ishigami immersed himself in the problem and soon lost track of time. He was an explorer heading out on safari, a soldier diving headlong into battle, an engine fueled by excitement and pride. His eyes never left the formulas for a moment, his every brain cell devoted to their manipulation.
* * *
“Ah!” Ishigami stood suddenly. Report paper in hand, he whirled around. Yukawa had put on his coat and was curled up in a ball on the floor, sleeping. Ishigami walked over, stopped, and shook his shoulder. “I figured it out.”
Yukawa sat up, eyes bleary with sleep. He rubbed his face and looked up at Ishigami. “What’s that?”
“I figured it out. I’m sorry to report that this counterexample is wrong. It was an interesting approach, but there was a fundamental flaw in the distribution of prime numbers—”
“Hold on a second. Hold on.” Yukawa held up his hands. “My brain is nowhere near awake enough to understand whatever you just said. I’m not sure I would understand even after a few cups of coffee, for that matter. To be honest, I don’t know the Riemann hypothesis from a hole in the ground. I just brought that because I thought you’d be interested.”
“But you said you thought it was heading in the right direction?”
“I was just repeating what the professor over in the mathematics department said. Actually, he knew about the flaw in his counterproof. That’s why he didn’t publish it.”
“Oh. No wonder I found it,” Ishigami said, crestfallen.
“No, it is a wonder. I’m impressed. The professor told me that even a top-flight mathematician would never find that error in one sitting.” Yukawa looked at his watch. “And you did it in only … six hours. Impressive!”
“Six hours?” Ishigami looked out the window. The sky was already whitening. He glanced at the alarm clock to see that it was almost five A.M.
“Ishigami the Buddha lives on!” Yukawa cheered. “Some things never change. Which is kind of a relief.”
“Sorry, Yukawa. I totally forgot you were still here.”
“Oh, I don’t mind. Still, you should probably get some sleep. You’ve got school, don’t you?”
“That I do. But now I’m too excited to sleep. I haven’t concentrated on something like this for a long time. Thank you.” Ishigami extended a hand.
“I’m glad I came,” Yukawa said, giving him a firm shake.
“Me, too.” Ishigami nodded. “There’s not much to do here, but feel free to make yourself at home until the trains start running again.”
* * *
Ishigami slept until seven o’clock. He slept deeply, either because his brain was tired or from a deep psychological satisfaction, and when he woke his mind was unusually clear.
He was bustling about, getting ready for work, when Yukawa commented, “Your neighbor was up early.”
“My neighbor?”
“I just heard them leaving. A little after six thirty, I guess.”
So Yukawa had been awake.
Ishigami was wondering if he should say something when Yukawa continued, “That Detective Kusanagi I was telling you about says that she’s a suspect. That’s why he dropped in on you, isn’t it?”
Ishigami assumed an air of calm and put on his suit jacket. “He tells you about his cases, does he?”
“Now and then. It’s more like him dropping by to shoot the breeze, then complaining to me about work before I can get rid of him.”
“I still don’t know what the whole thing is about. Detective … Kusanagi, was it? He didn’t give me any details.”
“Well, apparently, a man was murdered. Your neighbor’s ex-husband.”
“Huh. Never would have guessed it,” Ishigami said, his face expressionless.
“You talk to your neighbor much?” Yukawa asked.
Ishigami’s brain went into overdrive. Judging from the tone of his voice, Yukawa didn’t suspect anything. That wasn’t why he was asking questions. Simply brushing him off was an option here. But Yukawa knew the detective—he had to consider that. Yukawa might mention his visit here. Ishigami had to answer.
“I wouldn’t say ‘much,’ but I do frequent the lunch box shop where Ms. Hanaoka—that’s her name—works. Forgot to mention that to Detective Kusanagi, now that I think about it.”
“So, she’s a seller of lunch boxes,” Yukawa mused.
“I don’t go there because my neighbor works there—she just happens to work at the store where I buy my lunch, if you follow. It’s near the school.”
“I hear you. Still, I can’t imagine it’s all that pleasant having a murder suspect in the neighborhood.”
“It wasn’t me she murdered, so I don’t see how it’s any of my business.”
“How very true,” Yukawa said, without a shred of suspicion.
They left the apartment at seven thirty. Yukawa decided not to head for the nearest station, instead saying he’d walk with Ishigami to his school and take the train from there, a route that would save him from having to make a transfer.
Yukawa didn’t speak of the case or Yasuko Hanaoka again. At first Ishigami had wondered if Kusanagi had sent him to get information, but now he decided he’d probably been overthinking the situation. Kusanagi would have
no reason to go to such lengths to get information from him anyway.
“Interesting commute you have here,” Yukawa commented. They crossed under the Shin-Ohashi Bridge and began to walk along the slow-flowing Sumida River, past the ramshackle village set up by the homeless.
The gray-haired man with the ponytail was hanging up his laundry. Beyond him, the Can Man was well into his daily routine.
“It’s the same thing every day,” Ishigami said. “This entire past month, nothing’s changed a bit. You could set your watch by these people.”
“That’s what happens when you free people from the restraints of time. They make their own rigid schedule.”
“I couldn’t agree more.”
They went up the stairs just before Kiyosu Bridge, in the shadow of a nearby office building. Seeing their reflection in a glass door on the first floor, Ishigami shook his head. “How have you managed to stay so young, Yukawa? You still have a full head of hair. How different we two are!”
“Not as full as it used to be. And what’s underneath it is slowing down, too.”
“Good thing it was going too fast to begin with.”
While they chatted, Ishigami felt himself growing tense. If they kept on like this, Yukawa would come with him all the way to Benten-tei. He started to worry that this genius observer of the natural world might notice something between him and Yasuko Hanaoka if he happened to see the two of them together. And he didn’t want to fluster Yasuko by suddenly arriving with a stranger.
When he saw the sign he pointed it out. “There is the lunch box place I was telling you about.”
“Benten-tei, huh? Interesting name. The owners must have hoped that Benten, the goddess of wealth, would smile down on them.”
“Well, they have my business. I’ll be buying one there again today.”
“Right. Well, I suppose I’d best be off, then.” Yukawa stopped.
This was unexpected—and welcome, Ishigami thought. “I’m sorry I wasn’t the best host.”
“Not at all. You sell yourself short.” Yukawa narrowed his eyes. “You ever think about going back to the university to continue your research?”
Ishigami shook his head. “Anything I can do at a university I can do on my own. And I doubt any place would be willing to take me at my age.”
“You might be surprised, but I won’t twist your arm. Good luck, Ishigami.”
“You, too, Yukawa.”
“It was good seeing you again.”
The two shook hands, and Ishigami stood on the sidewalk to see his friend off. He wasn’t being sentimental. He just didn’t want Yukawa to see him go into Benten-tei.
When Yukawa had disappeared into the distance, Ishigami turned and walked swiftly toward the shop.
SEVEN
Yasuko breathed a sigh of relief when she saw Ishigami’s face. He looked calm, and that made her calm somehow. She had noticed that he had a visitor the night before and that the lights in his apartment were on late into the night. She was afraid that the detective had come back to question him again.
“I’ll take the special.” He placed the order in his usual monotone. As always, he did not so much as glance up at Yasuko.
“One special, coming up. Thank you,” she said. Then she added in a low voice, “You had a visitor yesterday?”
Ishigami stammered, then glanced up at her, blinking. He looked around the room quickly. “We shouldn’t speak like this,” he whispered. “They might be watching.”
“I’m sorry,” Yasuko said, withdrawing behind the counter.
The two said nothing while they waited for the lunch to be ready. They didn’t even exchange glances.
Yasuko glanced out at the street but couldn’t see anyone watching. Of course, if the police really were out there, they wouldn’t be so obvious about it.
The boxed lunch arrived from the back. She handed it to Ishigami.
“He was an old classmate,” Ishigami said abruptly as he was paying.
“What?”
“An old classmate from university came to visit. I’m sorry if it alarmed you.” Ishigami talked without moving his lips.
“Oh, no … I’m fine,” Yasuko said, smiling despite herself. Then she looked down at the floor, so that anyone watching from outside would not see her mouth. “I see. I wondered—you don’t seem to have many visitors.”
“He was my first, actually. It was a surprise to me.”
“Well, I’m glad for you, then.”
“Right, thanks,” Ishigami said awkwardly as he took the bag with his lunch in it from her. “Tonight, then.”
Which meant he would be calling her again this evening. Yasuko nodded, smiling.
She watched the back of his stocky figure as he stepped out onto the street, wondering that even a man who was as much of a hermit as he was should have friends.
When the morning rush had subsided, she went to take her usual break in the back of the shop with Sayoko and her husband. Sayoko had a sweet tooth; she put out a couple of sweet bean cakes for her and Yasuko. Mr. Yonazawa, whose tastes ran more toward the salty and savory, looked at the cakes without interest and sipped at his tea. The part-timer, Kaneko, was out making deliveries.
“Did they bug you again yesterday?” Sayoko asked after a sip of tea.
“Who?”
“Them. The cops.” Sayoko frowned. “They had no end of questions for us during the day, you know. I figured they might drop by your place again at night. That’s what we thought,” she said, looking to her husband. The taciturn Yonazawa merely nodded.
“No, no. Nothing since that time I told you about.”
Actually, Misato had been questioned outside her school, but Yasuko didn’t see the need to tell them that.
“Well then, I’m glad. They can be very persistent.”
“All they were doing was asking a few questions,” Yonazawa said. “It’s not like they really suspect Yasuko of anything. They just have to follow procedure.”
“I suppose the police are public servants, after all. Still, even if she isn’t a suspect, I’m sure glad that her ex didn’t come here. If he had stopped in before he was killed, then the cops would be after Yasuko something fierce.”
“Now don’t go scaring her with your foolishness,” Yonazawa scoffed at his wife.
“You never know! Wasn’t the detective saying that Togashi had been asking around at Marian for Yasuko?” Marian was the club in Kinshicho where Yasuko and Sayoko had worked. “He said it would have made sense for him to come here next. He was suspicious of something. I saw it on his face.”
“Well, maybe—but that Togashi never did show up here, so we don’t have to worry about it, do we?”
“That’s why I said I’m glad he didn’t. If he had, then mark my words, the poor girl wouldn’t get a moment’s peace.”
Yonazawa frowned and shook his head, unconcerned. Yasuko wondered how he’d react if he found out that Togashi really had come to the shop. She felt a knot form in her stomach.
“Well, either way, it can’t be pleasant. You just need to hang in there, Yasuko,” Sayoko said brightly. “Of course the police will be investigating you; he was your ex-husband, after all. But once they are satisfied you had nothing to do with it, they’ll leave you be. And then you’ll be really free at last. I know Togashi was still hanging around your neck.”
Yasuko forced herself to smile.
“To be perfectly honest, I’m glad he got himself killed,” Sayoko said emphatically.
“Hey now.” Yonazawa frowned.
“Oh, what’s the harm in telling it like it is? You just don’t know how much trouble poor Yasuko went through on that man’s account.”
“And you do?”
“Well, I have an idea—Yasuko’s told me quite a bit. She started working at Marian just to get away from him, isn’t that true? And when I think about him snooping around there looking for her—why, it just gives me the shivers. I don’t know who killed him, but I’d like to shake their hand.??
?
Yonazawa rolled his eyes and stood. Sayoko, looking disgruntled, watched him leave, then turned to Yasuko. “Still, you have to wonder what did happen. Maybe some loan shark finally caught up with him?”
“Who knows?” Yasuko shrugged.
“Well, I hope you’re spared the worst of it. That’s all I’m concerned about,” Sayoko said quickly, picking up the last bit of bean cake and popping it in her mouth.
Yasuko returned to the front of the shop, the conversation weighing on her spirit. It was clear that the Yonazawas didn’t suspect anything. To the contrary, they were concerned that the whole investigation was going to be a nuisance for her. It hurt to deceive them, but they would be in for far more trouble if she were arrested. That would be sure to hurt business at Benten-tei. She really had no other choice but to continue to conceal the truth from them.
She went about her work mechanically as the afternoon wore on, haunted by thoughts of her predicament. Each time her mind began to wander, she forced herself to focus on the task at hand, concentrating on the customers as best she could.
It was approaching six o’clock, and no customers had been in for a while, when the shop door swung open.
“Welcome,” Yasuko said reflexively. Then she glanced at the new arrival. Her eyes widened. “My…”
“Heya.” The man who stood before her smiled, wrinkles forming at the corners of his eyes.
“Mr. Kudo!” Yasuko put her hands to her open mouth. “What are you doing here?”
“What do you mean? I’m here to buy a lunch box,” Kudo said, looking up at the row of pictures showing the various boxed lunches they had for sale. “Say, pretty nice selection you got here.”
“They told you where to find me at Marian.”
“Something like that.” He grinned. “I dropped by there the other day. First time in a while.”
Yasuko called into the back of the shop, “Sayoko! Come quick! You’re never going to believe this!”
“What is it? What’s wrong?” Sayoko asked, rushing out.