The Pair That Won't Quit

The bus smelled like wet coats and the heater that only worked in the back. Maya had her finger on the window, wiping a clear stripe through the fog.

"Five and seven," she said. "Eleven and thirteen. Seventeen and nineteen."

"You're skipping numbers," Soren said.

"On purpose. Those are the ones where both of them are prime and they're only two apart. Twins." She breathed on the glass and it went white again. "I want to know how far they go."

Soren already had the notebook open on his knee. He wrote a column of primes, the ones he could do in his head. Two, three, five, seven, eleven, thirteen, seventeen, nineteen, twenty-three.

"They thin out," he said. "Look. Down here at the start they're packed. By the time you get past a hundred there are gaps. By a thousand the gaps are huge."

"Right. So why do the twins keep showing up?"

The bus hit a bump and the trophy rattled in the aisle. Second place. Neither of them had mentioned it for an hour.

Soren did the thing he did, which was test it. He picked a number neighborhood far from the small cozy ones. "Okay. Around a hundred. Is there a twin pair?"

Maya stared at the seat in front of her. "Hundred one and a hundred three."

"Are those both prime?"

"I think so. Hundred one isn't even, doesn't end in five, three doesn't go in, seven doesn't, eleven doesn't." She counted on her fingers under the seat. "Yeah. And a hundred three, same."

Soren wrote them down. "So even up there, where it's getting empty, there's still a pair. Two apart."

"Go further," Maya said. "Make it harder."

He picked a bigger number, and it took longer, and they got one wrong and had to back up. But they found a pair again. Then a stretch with none at all, a long bare patch, and Maya's stomach dropped a little, like the twins had finally quit.

"There," she said suddenly, further along. "There's another one."

"You can't see that. You don't know those are prime."

"I don't know. I'm guessing they're there. They keep being there." She rubbed the fog away again. "Every time we think the gaps got too big, two of them are still holding hands."

Soren chewed his pen. "That's the part I don't get. The primes spread out. That's not a maybe. That's proven, you can prove they get rarer forever. So you'd think the chance of two landing right next to each other would basically die out."

"But it doesn't."

"It doesn't look like it does. We keep finding pairs."

Maya turned around in her seat to face him fully, which she only did when something had her by the collar. "So has somebody proved it? That the pairs go forever?"

Soren looked at his column of numbers. He was honest about what he didn't know, always, even when it cost him.

"I don't think so," he said slowly. "I've read about it. People have checked further than you could write down in a whole life. Trillions and trillions. The twins keep coming. Computers find them way out where the numbers are gigantic."

"So it's true."

"No. That's the thing." He set the pen down. "Finding them isn't the same as proving they never stop. You could check a billion pairs and there could still be a last one, somewhere out past where anyone ever looked. Nobody has been able to show that there isn't a last pair."

Maya went quiet. The bus windows had gone full dark outside now, just the orange smear of highway lights sliding by.

"Wait," she said. "Wait. So this isn't a homework thing. This isn't a thing where the teacher knows the answer and we're catching up."

"No."

"Nobody knows."

"Nobody on Earth knows," Soren said. "They have a name for it. The twin prime conjecture. Conjecture means we think it's true and we can't prove it."

Maya put both hands flat on her knees. "How long."

"What?"

"How long have people not known."

Soren thought. "A long time. Way more than a hundred years. People way smarter than us, with their whole lives, and the question's just been sitting there. Open."

The heater rattled. Somebody up front laughed at a phone.

Maya felt something she didn't have a word for. It was the feeling of a room she'd thought was small turning out to have no back wall. She had always been the kid who asked one question too many, the one teachers answered and then looked relieved when she stopped. And here was a question that the answer-people, all of them, every grown expert with a desk and a degree, had run straight into and bounced off.

"They're still out there," she said. "The pairs. Right now. Past where anybody's checked."

"Probably," Soren said. "We can't prove it."

"But you think they are."

He didn't answer right away. He looked at his notebook, at the twins he'd found himself tonight, hundred one and hundred three, holding hands in his own handwriting.

"Yeah," he said. "I think they go forever. I just can't prove I'm right. And that's allowed. That's a real place you're allowed to stand."

Maya laughed, a short surprised one. "You're allowed to stand there."

"You're allowed to stand there for a hundred years if you have to."

She turned back to the window. She wiped a fresh stripe and started writing in the fog with one fingertip, the numbers reversed so they'd read right from outside, where nobody was.

Eleven. Thirteen.

A gap.

Seventeen. Nineteen.

"Keep going," Soren said.

She kept going. She wrote them bigger than she could verify, pairs she was only guessing at, out past where the glass was clear, into the part that was still fogged, and her finger left two clean tracks every time, two and two and two, all the way to the edge of the window.

The bus drove on into the dark, and the heater rattled, and on the cold glass the last pair she'd written was already beginning to bleed back into fog. Soren leaned across the aisle and wiped a clear spot just past it, and waited, with the pen, for her to fill it.