Factoring Humanity

This Integer	Divides Into 2,832 This Many Times
1	2,832
2	1,416
4	708
6	472
8	354
12	236
16	177
24	118
48	59
118	24
177	16
236	12
354	8
472	6
708	4
1,416	2
2,832	1

Of course, the assumption by most researchers was that there were 2,832 individual pages of data — but there might be as few as one page, made up of 2,832 tiles. Or there could be two pages, each made up of 1,416 tiles. Or three, made up of 944 tiles. And so on.

But how to tell which combination the Centaurs had intended?

She stared at the list, noting its symmetry: the first line was 1 and 2,832; the last was the reverse — 2,832 and 1. And so the lines were paired up and down until the middle two: 48 and 59; 59 and 48.

It was almost as if the middle two were the pivot, the axle on which the great propeller of figures rotated.

And—

Christ—

Except for 1, 3, and 177, the number 59 was the only possibly prime number on the list: all the others were even numbers and, by definition, couldn’t be primes.

And — wait. Kyle had taught her a trick years ago. If the digits composing a number added up to a number divisible by three, then the original number was also divisible by three. Well, the digits making up 177 — one, seven, and seven — added up to fifteen, and three went into fifteen five times, meaning 177 couldn’t be prime.

But what about the number 59? Heather had no idea how to determine if a number was prime, except by brute force. She made another quickie spreadsheet, this one dividing 59 by every whole number smaller than itself.

But none of them divided evenly.

None, except 1 and 59.

Fifty-nine was a prime number.

And — a thought occurred to her. One itself was sometimes considered a prime. Two was definitely a prime. So was three. But in a way, all those numbers were trivial primes: every whole number lower than them was also divisible only by itself or one. In many ways, five was the first interesting prime number — it was the first one in sequence that had numbers lower than itself that weren’t primes.

So if you discounted one, two, and three as trivially prime, then in the table she’d produced, 59 was the only non-trivial prime that divided evenly into the total number of undecoded alien messages.

It was another arrow pointing at that figure. The alien transmissions could possibly be arrayed in 48 pages each consisting of 59 individual messages, or 59 pages each consisting of 48 messages.

Researchers had been looking for recurring patterns in the messages for years, but so far none had turned up that hadn’t seemed coincidental. Now, though, that they knew the total number of messages, all sorts of fresh analyses could be done.

She opened another window on her computer and brought up the file directory of alien messages. She copied the directory into a text file, where she could play with it. She highlighted the bit counts for the first 48 undecoded messages and tallied them up: they totaled 2,245,124 bits. She then highlighted the next twenty-four. The tally came to 1,999,642.

Damn.

She then highlighted the counts for messages 12 through 71 — the first 59 undeciphered messages.

The total came to 11,543,124 bits.

Then she highlighted messages 72 to 131 and tallied their sizes.

The total was also 11,543,124 bits.

Heather felt her heart pounding; perhaps someone had noticed this before, but. . .

She did it again, working her way through the material.

Her spirits fell when she found the fourth group tallied only 11,002,997 bits. But after a moment, she realized she’d highlighted only 58 messages instead of 59. She tried again.

The tally was 11,543,124.

She continued on until she’d done all 48 groupings of 59 messages.

Each group totaled precisely 11,543,124 bits.

She let out a great whoop! of excitement. Fortunately, her office did have that sturdy oak door.

The aliens hadn’t sent 2,832 separate messages — rather, they’d sent 48 large ones.

Now, if only she could figure out how to tile the messages together. Unfortunately, they were of many different sizes, and there was no orderly repetition from page to page. The first message making up the first group of 48 was 118,301 bits long (the product of the primes 281 and 421), whereas the first message of page two was 174,269 bits long (the product of the primes 229 and 761).

Presumably, the individual tiles formed square or rectangular shapes when properly placed together. She doubted she could figure it out by trial and error.

But surely Kyle could write her a computer program that would do it.

After last night, she was hesitant. What would she say to him?

She steeled her courage and picked up her phone.

“Hello?” said Kyle’s voice.

He doubtless knew it was Heather calling; he could read it off the status line on his phone. But there was no particular warmth in his voice.

“Hi, Kyle,” said Heather. “I need your help.”

Frosty: “You didn’t need my help last night.”