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.
Which was why someone had proposed the exasperating P =
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.
Which was why someone had proposed the exasperating P =
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