An Elegant Solution(57)

“The sum is infinite.”

“But they also grow infinitely small?”

“But for this series, not as quickly. Not as quickly as the sum grows infinite.”

“That is correct.”

Oh, he had my interest piqued entirely. I thought he might really be a show-man, the way he drew out a puzzle and pulled his student into it.

“How do you know that the first series added to exactly one?”

“There is a method,” I said. This was still all very simple. He knew the method, of course. “If the series is multiplied by two, it becomes one, one half, one fourth, one eighth, and on. If the two are subtracted, the infinity of terms is cancelled, and the remaining value is one. So, the series subtracted from twice the series is the value of the series, which is one.”

“Does this method work for any infinite series?”

“No, sir. Only for this geometric type.”

“Now write these numbers.”

Then I knew, from his voice, that this was to be the challenge. The first questions had only been to set his stage, and now he was ready to play his drama. “One, one fourth, one ninth, one sixteenth, one twenty-fifth, one thirty-sixth, and on. What are these?”

¹⁄₁ + ¹⁄₄ + ¹⁄₉ + ¹⁄₁₆ + ¹⁄₂₅ + ¹⁄₃₆ + . . .

“They are one over the square of one, one over the square of two, one over the square of three, one over the square of four, and on. They are Reciprocal Squares.”

“Yes, Reciprocal Squares.” It was roast veal and wine, exquisite, the way he said it. “They are Reciprocal Squares. Is the sum infinite? Or finite?”

“Finite,” I said, though I paused to think. “Yes. Finite. Besides the beginning one, the numbers are each smaller than the first series you listed. One fourth is smaller than one half, one ninth is smaller than one fourth, one sixteenth is smaller than one eighth, and on. So if the first sum was finite, this must be also.”

“Yes. Finite. Very good, though that was simple.” And he paused, and his pause was perfect in length and depth and width. “And what is that finite sum of the infinite Reciprocal Squares?” he asked.

“The sum . . .” I was bewildered. I stared at the numbers on my paper and tried to make sense. I looked at the pattern of them, at what they seemed to be adding to, at the other methods I knew, anything. They seemed very simple, as simple as the other sums we’d done. And finally I grasped that . . . “I don’t know.” I realized minutes had gone by. “What is the sum?”

“What is it?” He rubbed his hands. “No one knows. It is a number, somewhat larger than one and a half, and less than two. It has been calculated to a close value. But no one knows what it really is. Perhaps it’s no particular number at all, just a number. But it should be something more important than that. A squared root, a cubed root, a ratio of important numbers. No one knows.”

“It would be something . . . surprising,” I said.

“Perhaps. Perhaps. And now, the Paris Academy has issued a challenge to anyone in Europe who might discover the true value of the Reciprocal Squares.” He’d kept the paper in his hand closed from me; now he opened it. “Monsieur Fontenelle and Monsieur de Molieres of the Academy are very great Mathematicians. I have instructed them myself. Their challenge is to all Europe.”

I was reading. The page was in Latin, of course, and it was just as he’d said. The two men were members and directors of the Royal French Academy, and their announcement was as weighty as a mountain: to explain the meaning of the Sum of the Infinite Reciprocal Squares.

Master Johann was a member of the Royal Academy; if he were not, the Academy would hardly have been worth anyone else’s membership. Therefore, he had received the first copy of the challenge, and soon it would go out to all the rest of Europe. And whoever first solved it, or proved it unsolvable, would instantly leap to the highest rank of Mathematicians, if he wasn’t already there. And I was being given this glimpse into their world.

“Will you try?” I asked. I had finally returned to the dim room in Master Johann’s house. He was waiting for me.

“I have reason to believe it can be expressed in some other way than only as the sum. It has some special value as itself.”

“Why?”

I saw in his eyes a look I’d seen sometimes before, which was like the Basel Walls when their gates were closed and a banner of plague or war was flying, to tell travelers that there was no entrance to the city, and they should keep a distance if they didn’t want an arrow for warning. “I will leave you to explore that.”