"Let's try finding the prime numbers up to 100," the Professor said one day when Root had finished his homework. He took his pencil and began making a list: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
It always amazed me how easily numbers seemed to flow from the Professor, at any time, under any circumstances. How could these trembling hands, which could barely turn on the microwave, make such precise numbers of all shapes and sizes?
I also liked the way he wrote his numbers with his little stub of a pencil. The 4 was so round it looked like a knot of ribbon, and the 5 was leaning so far forward it seemed about to tip over. They weren't lined up very neatly, but they all had a certain personality. The Professor's lifelong affection for numbers could be seen in every figure he wrote.
"So, what do you see?" He tended to begin with this sort of general question.
"They're scattered all over the place." Root usually answered first. "And 2 is the only one that's even." For some reason, he always noticed the odd man out.
"You're right. Two is the only even prime. It's the leadoff batter for the infinite team of prime numbers after it."
"That must be awfully lonely," said Root.
"Don't worry," said the Professor. "If it gets lonely, it has lots of company with the other even numbers."
"But some of them come in pairs, like 17 and 19, and 41 and 43," I said, not wanting to be shown up by Root.
"A very astute observation," said the Professor. "Those are known as 'twin primes.' "
I wondered why ordinary words seemed so exotic when they were used in relation to numbers. Amicable numbers or twin primes had a precise quality about them, and yet they sounded as though they'd been taken straight out of a poem. In my mind, the twins had matching outfits and stood holding hands as they waited in the number line.
"As the numbers get bigger, the distance between primes increases as well, and it becomes more difficult to find twins. So we don't know yet whether twin primes are infinite the way prime numbers themselves are." As he spoke, the Professor circled the consecutive pairs.
Among the many things that made the Professor an excellent teacher was the fact that he wasn't afraid to say "we don't know." For the Professor, there was no shame in admitting you didn't have the answer, it was a necessary step toward the truth. It was as important to teach us about the unknown or the unknowable as it was to teach us what had already been safely proven.
"If numbers never end, then there should always be more twins, right?"
"That makes sense, Root. But when you get to much bigger numbers—a million or ten million—you're venturing into a wasteland where the primes are terribly far apart."
"A wasteland?"
"That's right, a desert. No matter how far you go, you don't find any. Just sand as far as the eye can see. The sun shines down mercilessly, your throat is parched, your eyes glaze over. Then you think you see one, a prime number at last, and you go running toward it—only to find that it's just a mirage, nothing but hot wind. Still, you refuse to give up, staggering on step by step, determined to continue the search ... until you see it at last, the oasis of another prime number, a place of rest and cool, clear water...."
The rays of the setting sun stretched far into the room. Root traced the circles around the twin primes as the steam from the rice cooker floated in from the kitchen. The Professor stared through the window as if he were looking out at the desert, though all he could really see was his tiny, neglected garden.
The thing the Professor hated most in the whole world was a crowd, which is why he was so reluctant to leave the house. Stations, trains, department stores, movie theaters, shopping malls—any place people gathered in large numbers was unbearable for him. There was something fundamentally incompatible between crushing, random crowds and pure mathematical beauty.
The Professor wanted peace, though that didn't necessarily mean complete silence. Apparently, he was not disturbed by Root when he ran down the hall or turned up the volume on the radio. What he needed was internal calm uninterrupted by the outside world.
When he had solved a contest problem from one of his journals and was making a clean copy to put in the mail, you could often hear him murmur, "How peaceful ..." He seemed to be perfectly calm in these moments, as though everything were in its rightful place, with nothing left to add or subtract. "Peaceful" was, to him, the highest compliment.
When he was in a good mood, he would sit at the kitchen table and watch me making dinner; and if I were making dumplings, he would look on with something approaching wonder. I would take a dumpling skin in the palm of my hand, spoon on a bit of filling, and then pinch up the edges before setting it on the platter. A simple process, but he was completely absorbed by it, watching me until the last dumpling had been stuffed. I have to admit that the scene struck me as so funny that I hardly could keep from laughing.
When I was done at last and the dumplings were neatly arranged on the plate, he would fold his hands on the table and nod solemnly. "How peaceful ..."
On the sixth of May, at the end of the spring holidays, Root cut himself with a kitchen knife. The Professor did not take it well.
After the four-day break, I arrived at the Professor's house only to discover that the sink had been leaking and a puddle had spread into the hall. By the time I'd called to have the water shut off and hired a plumber to come in, I was probably a bit out of sorts. To make matters worse, the Professor had seemed more remote than ever, and no matter how often I pointed out my picture among the tags on his coat, he seemed confused or oblivious. By evening he had still not come out of his shell. While my irritation might have contributed to Root's accident, the Professor was in no way to blame.
Shortly after Root arrived from school, I realized that I'd run out of cooking oil. I was uneasy leaving the Professor and Root alone, so I talked to Root before I left.
"Do you think it's okay?"
"Is what okay?" he replied, almost curtly. It is hard to say exactly what worried me, I had no premonition, I was simply anxious about leaving the Professor in charge.
"I've never left you alone with the Professor and I was just wondering if that's okay—"
"Don't worry!" Root said, running off to the study to have his homework checked.
I was gone no more than twenty minutes, but when I opened the door, I knew immediately that something was wrong. I discovered the Professor, sobbing and moaning, crouched on the kitchen floor, holding Root in his arms.
"Root ... Root ... his hand!"
He could barely speak, and the more he tried to explain what had happened, the more incoherent he became. His teeth chattered and sweat poured down his face. I pried Root loose from his arms.
Root wasn't crying. He may have been trying to keep the Professor calm, or he may have been afraid I would be angry with him, but whatever the reason, he had been lying quietly in the Professor's arms, waiting for me to return. Their clothes were smeared with blood and the cut on Root's hand was still bleeding, but I could see right away that the Professor's panic was out of all proportion to Root's injury. The bleeding had nearly stopped, and Root didn't appear to be in any pain. After I'd washed out the wound at the kitchen sink, I brought him a towel and told him to hold it on the cut. In the meantime, the Professor sat motionless on the floor, his arms frozen as if he were still holding Root. It seemed almost more urgent to look after him than it was to treat Root.
"Don't worry," I said, patting him gently on the back.
"How could this have happened? Such a sweet, good boy ..."
"It's just a little cut. Boys hurt themselves all the time."
"But it's all my fault. He didn't do anything wrong. He didn't want to bother me, so he didn't say anything ... he just sat there bleeding...."
"It's no one's fault," I said.
"No, it's my fault. I tried to stop the bleeding, but I couldn't.... And then he got so pale, and I was afraid he'd stop breathing...." He hid his
face in his hands, covering the sweat and tears.
"Don't worry," I said again. "He'll be fine." As I rubbed his back, I realized that it was surprisingly broad and sturdy.
Neither Root nor the Professor were making much sense, but I finally managed to piece together what had happened: Root had finished his homework and was trying to peel an apple for a snack when he had cut himself between his thumb and index finger. The Professor insisted that Root had asked him for help with the apple, while Root maintained that he'd done the whole thing by himself. In any event, Root had tried to take care of the cut but he couldn't stop the bleeding, and the Professor had found him just as he'd begun to panic.
Unfortunately, the clinics in the neighborhood had already closed for the day. The only doctor answering the phone was a pediatrician at a clinic behind the train station, who said he could see him right away. I helped the Professor up and dried his face, and at that point an astonishing change came over him. He hoisted Root onto his back, and though I tried to remind him that the child hadn't hurt his legs, he ran off to the doctor's carrying Root piggyback. To be honest, the ride seemed so rough that I was worried the wound would open up again. It could hardly have been easy for the Professor to carry a sixty-pound child on his back, but he was stronger than I'd thought. He charged along in his moldy shoes, gasping a bit from time to time, but holding Root's legs firmly under his arms. Root pulled his Tigers cap down over his eyes and buried his face in the Professor's back, less from pain than from the embarrassment of being seen. When we got to the clinic, the Professor pounded on the locked door, as though he were carrying a dying child on his back.
It took only two stitches to close the cut, but the Professor and I had to wait in the darkened corridor until they had finished the examination. They wanted to be sure Root hadn't severed a tendon.
The clinic was old and depressing. The ceiling was discolored, and the grimy slippers stuck to your feet. Yellowed posters on the walls gave instructions for weaning and inoculations. The only light in the hall was the dim bulb outside the X-ray room.
They'd said the test was just a precaution, but Root had been in the examination room for some time.
"Have you ever heard of triangular numbers?" the Professor said, pointing at the radiation sign on the door of the X-ray room. It was shaped like a triangle.
"No," I said. He sounded calm now, but I could tell that he was still a little shaken.
"They're truly elegant," he said, beginning to draw dots on the back of a questionnaire that he'd picked up in the lobby.
"What do you make of these?"
"Well, let's see. It looks like neatly stacked firewood, or maybe rows of beans."
"That's right, the point is they're 'neatly' arranged. One in the first row, two in the second, three in the third.... It's the simplest way to form a triangle." I glanced at the dots on the page. The Professor's hand was trembling slightly. The black marks seemed to float up in the half-light. "So then, if we total up the number of dots in each triangle, we get 1, 3, 6, 10, 15, and 21. And if we write these as equations:
1
1 + 2 = 3
1 + 2 + 3 = 6
1 + 2 + 3 + 4 = 10
1 + 2 + 3 + 4 + 5 = 15
1 + 2 + 3 + 4 + 5 + 6 = 21
"In other words, a triangular number is the sum of all the natural numbers between 1 and a certain number. Then, if you put two of these triangles together, things get even more interesting. Why don't we look at the fourth one, 10, so we don't have to draw too many dots?"
It wasn't particularly cold in the hall, but the trembling in his hand had grown worse and the dots had slightly smudged. His whole being seemed concentrated in the tip of his pencil. A few of the notes on his suit were smeared with blood and now illegible.
"Look at this. When you put two of the four-row triangles together, you get a rectangle that is 4 dots high and 5 dots wide; and the total number in the rectangle is 4 × 5 or 20 dots. Do you see that? And if you divide that in half, you get 20 ÷ 2 = 10, or the sum of the natural numbers from 1 to 4. Or, if you look at each line of the rectangle, you get:
"And once you know that, you can use this relationship to figure out the tenth triangle—the sum of the numbers from 1 to 10—or the hundredth or any other. For 1 to 10 it would be:
"And for 1 to 100,
"And 1 to 1000,
"And 1 to 10,000...."
The pencil rolled out of his hand and fell at his feet. The Professor was crying. I believe it was the first time I saw him in tears, but I had the feeling that I'd seen these emotions many times before. I placed my hand on his.
"Do you understand?" he said. "You can find the sums of all the natural numbers."
"I understand."
"Just by lining up the dots in a triangle. That's all there is to it."
"Yes, I see that now."
"But do you really understand?"
"Don't worry," I told him. "Everything's going to be all right. How can you cry, look at these beautiful triangular numbers."
Just then the door to the examination room opened and Root emerged.
"See!" he said, giving his bandaged hand a wave. "I'm fine."
Leaving the clinic, we suddenly realized that we were starving, so we decided to eat out. Since the Professor hated crowds, we went to the emptiest restaurant in the arcade near the station and had a bowl of curry and rice. There were almost no other customers, so we might have guessed that the curry wouldn't be particularly good; but Root, who almost never ate out, was delighted. He also seemed pleased to have such a dramatic bandage (for a relatively minor injury), as if he were the hero of some great battle.
"I won't be able to help with the dishes or even take a bath for a while," he said, a bit full of himself.
The Professor carried him home on his back, and Root was less worried about being seen now that it was dark. Perhaps he was just being considerate of the Professor. Whatever the reason, he climbed on without objection and rode happily. A thin sliver of moon hung above the row of sycamores glowing under the street lamps. A pleasant breeze was blowing, our stomachs were full, and Root's hand would heal. I felt a great sense of contentment. My footsteps fell in with the Professor's, and Root's tennis shoes swung back and forth in time.
After seeing the Professor home, we headed back to our apartment. For some reason Root was suddenly in a bad mood. He went straight to his room and turned on the radio, and refused to answer when I called to tell him to take off his bloodstained clothes.
"Are the Tigers losing?" I asked. He was standing at his desk, glaring at the radio. They were playing the Giants. "They lost yesterday, didn't they?" Still no answer. The announcer informed us that the score was tied 2-2 in the bottom of the ninth, with Nakata and Kuwata locked in a pitchers' duel. "Does it hurt?" I asked. He bit his lip and kept his eyes on the radio. "If it hurts, take the medicine the doctor gave us. I'll get you some water."
"No," he said.
"You really should," I coaxed. "You don't want it to get infected." "I said it doesn't hurt."
He clenched his bandaged fist and rapped it against the desk, using his good hand to hide the tears welling up in his eyes. This clearly had nothing to do with the Tigers.
"Why are you doing that?" I said. "They just finished stitching you up. What am I supposed to do if you start bleeding again?"
Tears streamed down his cheeks now. I tried to check whether blood was soaking through the bandage, but he brushed my hand away. Cheers erupted from the radio—a two out single.
"Are you mad because I went out and left you with the Professor? Or are you embarrassed because you couldn't handle the knife? Or because you made a mistake in front of the Professor?" He'd fallen silent again. Kameyama was up at bat.
"Kuwata has been nearly unhittable.... He's struck out his last two at bats ... will it be another fastball? ... Here's the windup...."
The cheers rose again and again, drowning out the announcer, but Root seemed indifferent. He sat perfectly still
as the tears continued to roll down his cheeks.
I realized I had seen two men cry this evening. I had, of course, seen Root's tears countless times before—as an infant, when he'd wanted to be held or fed; and later, during tantrums, or when he lost his grandmother. And, for that matter, at the moment he came into this world. But these tears were different, and no matter how I tried to wipe them away, they seemed to flow from a place I could never reach.
"Are you mad because the Professor couldn't dress the wound properly?" I asked at last.
"No," said Root. He stared at me for a moment and then he spoke so calmly it seemed as though he had completely regained control of himself. "I'm mad because you didn't trust him. I'll never forgive you for that."
Kameyama hit the second pitch into right center, and Wada scored from first to end the game. The announcer was shouting and the roar of the crowd swept over us.
The next day, the Professor and I recopied his note tags. "I wonder where all this blood came from," he said, checking himself for a cut.
"Root, my son, hurt his hand with a knife—but it wasn't serious."
"Your son? That's terrible! It looks like it bled all over."
"No, he's just fine, thanks to you."
"Really? I helped?"
"Of course. How do you think the blood got on you?"
I pulled the notes from his suit one by one. Most were covered with an incomprehensible scrawl of math symbols—as though few things other than numbers were worth remembering.
"And when you finished helping Root, you taught me something important in the waiting room."
"Something important?"
"You taught me about triangular numbers, and the formula for finding the sum of the natural numbers from 1 to 10—something I could never have imagined, something sublime." I held out the most important note. "Shall we start with this one?"