The Plane

more than ever before.

“I solved it!”

“Huh?” Cold blinked. “Really?”

“Yeah. Can you go back to the puzzle?”

Cold stood up, and did as she was told. “What is the answer?”

“It's infinite.”

“Can you explain why?”

“I could... do you think it's necessary?”

Cold hesitated. Time was valuable, and if she was certain, they wouldn't receive any benefit from her knowing it too. But even so, she wasn't ready to gamble her life on something she didn't understand.

“If something happens and we have to hurry, you can submit the answer. But otherwise... I'd like to know.”

“I understand. So... okay. There are three parts to this proof. Reminder, the set holds numbers which can only be divided by one or themselves. Part one is, no number except one is a factor of two consecutive numbers, because if it's a factor of the original number, then multiplying it by even just one more will already result in a result greater than number next in size. Got it?”

“Yes,” Cold said after a few moments.

“Part two is, every number can be split into a product of only members of the set, because... well, let's start with any one number. Either it is already in the set, or it can be split into a product of two other numbers which are not one or itself, because that is the definition of numbers outside of the set. So, now we got two numbers which are smaller than our original one, and we can do the same for both of them. We can keep doing that until we're only left with numbers from the set. They get smaller every step, so we have to make it eventually.”

Cold had reached their destination by now, and sat down next to the two shapes. It took her several minutes to fully process this, but eventually she nodded. “Yes.”

“Good, that was the hardest part. Part three is... let's say we have any amount of numbers from the set. We can create the product of all of these numbers, resulting in a new number which is extremely high, and has all of our original numbers  as factors, right?”

Cold squeezed her eyes, but she was still following. “Yeah.”

“Good, next we increment this number by one. Now it is still insanely high, but it doesn't have any of our members as factors, based on part one, right?”

“Yeah.”

“So, and now we split it like in part two, until it is only made of numbers which are in the set. It may have already been in the set, in that case we get just one new number, or it