The Number That Kept Its Promise

The rain had trapped them inside, so Maya was feeding numbers to the companion just to watch it think.

The companion was a flat gray tile the school lent out, no bigger than a coaster. Soren had it propped against the sugar jar. You could ask it to test almost anything, and it would check millions of cases before you finished blinking.

"Pick a number," Maya said. "Any whole number. If it's even, cut it in half. If it's odd, triple it and add one. Keep going."

Soren tried seven. Seven became twenty-two, became eleven, became thirty-four, then seventeen, then fifty-two, then twenty-six, then thirteen, down and down, bouncing up sometimes, until it dropped to one and stuck.

"It always lands on one," Maya said. "I read about it. Always."

"Always for seven," Soren said. He liked to know where the edges of a word were. "Companion, test it."

The tile warmed under his fingers. Test target, it said in its calm voice. The Collatz pattern. Range: one through one billion.

Numbers blurred across the little screen too fast to read. It finished in under a second.

All cases reach one, the companion said. Pattern holds. Status: unresolved.

Maya frowned. "That's a contradiction. It held. It just said it held."

"It didn't say true," Soren said. He pulled his notebook closer. "It said unresolved. That's a different word."

"Make it go bigger," Maya said.

Soren did. One through one trillion. The tile took longer this time, almost three whole seconds, and the kitchen clock ticked through them.

All cases reach one, it said again. Status: unresolved.

Maya leaned back. "A trillion. Every single one came home. And it still won't say true."

"Companion," Soren said. "What does unresolved mean?"

The tile was quiet for a moment, the way it got when the answer wasn't a number.

I have checked every case I was given, it said. In all of them the pattern holds. I have not been given a reason it must hold for every number that exists. There are infinitely many I have not checked. Unresolved means: no one has proven it cannot fail.

Maya went still. That was the stillness Soren knew. The one before she said something he hadn't thought of yet.

"Wait," she said. "It can count forever and it still wouldn't be enough?"

"There's always a next number," Soren said slowly. "However high it counts. There's one more after."

He wrote that down. There is always a next one. Then he stopped, his pen above the page, because the size of that suddenly arrived in him all at once.

"So it could check a trillion," he said. "A trillion trillion. And somewhere out past all of that, past anywhere a machine could ever reach, one number could break the promise. And we'd never know, because we can't get there."

"Unless," Maya said.

"Unless what?"

She was tapping the table now, fast. "Unless you didn't have to get there. Unless you could prove it without checking each one. Like." She grabbed for an example. "Like how I know every even number can be cut in half. I didn't check them all. I just know what even means."

Soren felt the floor of the idea drop away beneath him.

"That's it," he said. "That's the difference. The companion is checking. A proof isn't checking. A proof is a reason."

"A reason that covers numbers you'll never even see," Maya said.

They looked at the gray tile sitting against the sugar jar. "Companion," Maya said. "Has anyone ever proven it? The reason. For this one."

No, the tile said. The Collatz conjecture has been tested far beyond a trillion. It has never failed. It has never been proven. It has remained unresolved for more than eighty years.

Eighty years. Soren thought about all the people in those eighty years, the cleverest people on Earth, sitting where he was sitting, watching the numbers come home to one over and over, and never once being able to say the word true.

"It's not that they weren't smart enough," Maya said quietly, and he could tell she was working it out as she spoke. "It's that nobody has found the reason yet. It might be small. It might be hiding in plain sight. It just hasn't been found."

"By anyone," Soren said.

"By anyone," Maya said.

And there it was, the thing that made the back of his neck go cold and bright at the same time. Every fact in his schoolbooks had a found feeling to it. Settled. Done by someone before he was born. But this was a question sitting open in the middle of the world, plain enough that an eleven-year-old could understand exactly what was being asked, and not one human being alive could answer it.

The rain pushed against the window.

"It feels weird," Maya said, "that the machine can be that sure and still not know."

"It's not unsure," Soren said. "It knows exactly what it knows. It knows it checked a trillion. It just won't pretend the trillion-and-first is checked when it isn't." He turned that over. "That's the most honest thing I've ever heard anybody say."

Maya almost laughed. "A coaster is the most honest thing you know."

"The most honest thing I know," he agreed.

She reached over and typed one more line. Not a billion. Not a trillion. She typed a number so long it ran off the edge of the little screen, a number neither of them could name, a number with no business existing except that someone had decided to write it.

The tile thought. The clock ticked. Outside a car hissed through a puddle.

Pattern holds, the companion said. Status: unresolved.

Maya pulled her knees up onto the chair and stared at the gray tile like it had just told her a secret it didn't know it was keeping.

"Somewhere out there," she said, "is the reason. Just sitting. Waiting for somebody to notice it."

Soren picked up his pen and drew an arrow pointing up, past the top of the page, off the paper, onto the wood of the table, and he didn't stop the line.