Yôko Ogawa - The Gift of Numbers aka The Housekeeper and the Professor

The white clippings of hair fell in clumps on the cape and then scattered to the floor. As he cut and combed away, did the barber suspect that the brain inside this snowy head could list all the prime numbers up to a hundred million? And did the customers on the sofa, waiting impatiently for the strange old man to depart, have any notion of the special bond between my birthday and the Professor's wristwatch? For some reason, I felt a secret pride in knowing these things, and I smiled at the Professor just a bit more brightly in the mirror.

After the barbershop, we sat on a bench in the park and drank a can of coffee. There was a sandbox nearby, and a fountain and some tennis courts. When the wind blew, the petals from the cherry trees floated around us and the dappled sunlight danced on the Professor's face. The notes on his jacket fluttered restlessly, and he stared down into the can as if he'd been given some mysterious potion.

"I was right-you look handsome, and more manly."

"That's quite enough of that," said the Professor. For once he smelled of shaving cream rather than of paper.

"What kind of mathematics did you study at the university?" I asked. I had little confidence that I would understand his answer; maybe I brought up the subject of numbers as a way of thanking him for coming out with me.

"It's sometimes called the 'Queen of Mathematics,' " he said, after taking a sip of his coffee. "Noble and beautiful, like a queen, but cruel as a demon. In other words, I studied the whole numbers we all know, 1, 2, 3, 4, 5, 6, 7… and the relationships between them."

His choice of the word queen surprised me-as if he were telling a fairy tale. We could hear the sound of a tennis ball bouncing in the distance. The joggers and bikers and mothers pushing strollers glanced at the Professor as they passed but then quickly looked away.

"You look for the relationships between them?"

"Yes, that's right. I uncovered propositions that existed out there long before we were born. It's like copying truths from God's notebook, though we aren't always sure where to find this notebook or when it will be open." As he said the words "out there," he gestured toward the distant point at which he stared when he was doing his "thinking."

"For example, when I was studying at Cambridge I worked on Artin's conjecture about cubic forms with whole-number coefficients. I used the 'circle method' and employed algebraic geometry, whole number theory, and the Diophantine equation. I was looking for a cubic form that didn't conform to the Artin conjecture… In the end, I found a proof that worked for a certain type of form under a specific set of conditions."

The Professor picked up a branch and began to scratch something in the dirt. There were numbers, and letters, and some mysterious symbols, all arranged in neat lines. I couldn't understand a word he had said, but there seemed to be great clarity in his reasoning, as if he were pushing through to a profound truth. The nervous old man I'd watched at the barbershop had disappeared, and his manner now was dignified. The withered stick gracefully carved the Professor's thoughts into the dry earth, and before long the lacy pattern of the formula was spread out at our feet.

"May I tell you about something I discovered?" I could hardly believe the words had come out of my mouth, but the Professor's hand fell still. Overcome by the beauty of his delicate patterns, perhaps I'd wanted to take part; and I was absolutely sure he would show great respect, even for the humblest discovery.

"The sum of the divisors of 28 is 28."

"Indeed…," he said. And there, next to his outline of the Artin conjecture, he wrote: 28 = 1 + 2 + 4 + 7 + 14. "A perfect number."

"Perfect number?" I murmured, savoring the sound of the words.

"The smallest perfect number is 6: 6 = 1 + 2 + 3."

"Oh! Then they're not so special after all."

"On the contrary, a number with this kind of perfection is rare indeed. After 28, the next one is 496: 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248. After that, you have 8,128; and the next one after that is 33,550,336. Then 8,589,869,056. The farther you go, the more difficult they are to find"-though he had easily followed the trail into the billions!

"Naturally, the sums of the divisors of numbers other than perfect numbers are either greater or less than the numbers themselves. When the sum is greater, it's called an 'abundant number,' and when it's less, it's a 'deficient number.' Marvelous names, don't you think? The divisors of 18- + 2 + 3 + 6 + 9-equal 21, so it's an abundant number. But 14 is deficient: 1 + 2 + 7 + 10."

I tried picturing 18 and 14, but now that I'd heard the Professor's explanation, they were no longer simply numbers. Eighteen secretly carried a heavy burden, while 14 fell mute in the face of its terrible lack.

"There are lots of deficient numbers that are just one larger than the sum of their divisors, but there are no abundant numbers that are just one smaller than the sum of theirs. Or rather, no one has ever found one."

"Why is that?"

"The answer is written in God's notebook," said the Professor.

Everything around us was glowing in the sunlight; even the dried shells of the insects floating in the fountain seemed to glitter. The most important of the Professor's notes-the one that read "My memory lasts only eighty minutes"-had come loose, and I reached over to adjust the clip.

"I'll show you one more thing about perfect numbers," he said, swinging the branch and drawing his legs under the bench to make more room on the ground. "You can express them as the sum of consecutive natural numbers."

6 = 1 + 2 + 3

28 = 1 + 2 + 3 + 4 + 5 + 6 + 7

496 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 + 27 + 28 + 29 + 30 + 31

The Professor reached out to complete the long equation. The numbers unfolded in a simple, straight line, polished and clean. The subtle formula for the Artin conjecture and the plain line of factors for the number 28 blended seamlessly, surrounding us where we sat on the bench. The figures became stitches in the elaborate pattern woven in the dirt. I sat utterly still, afraid I might accidentally erase part of the design. It seemed as though the secret of the universe had miraculously appeared right here at our feet, as though God's notebook had opened under our bench.

"Well then," the Professor said at last. "We should probably be getting home."

"Yes, we should," I said, nodding. "Root will be there soon."

"Root?"

"My son. He's ten years old. The top of his head is flat, so we call him Root."

"Is that so? You have a son? We can't dawdle then. You should be there when he gets home from school." With that, he stood to go.

Just then, there was a cry from the sandbox. A little girl stood sobbing, a toy shovel clutched in her hand. Instantly, the Professor was at her side, bending over to comfort her. He tenderly brushed the sand from her dress.

Suddenly, the child's mother appeared and pushed the Professor away, picking the girl up and practically running off with her. The Professor was left standing in the sandbox. I watched him from behind, unsure how to help. The cherry blossoms fluttered down, mingling with the numbers in the dirt.

"I did the problem and I got it right. So now you have to keep your promise and fix the radio." These were the first words out of Root's mouth as he came through the door. "Here, look," he said, holding out his math notebook.

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 55

The Professor studied Root's work as though it were a sophisticated proof. Unable to recall why he had assigned this problem or what connection it had to repairing the radio, he was perhaps looking for an answer in the sum itself.