Last week, in Number Sense 037, Awkward Goat proved that all counting numbers (numbers of goats in formation for the Marching Goat Society) were either prime numbers or the product of prime numbers (although he called them primary numbers, analogous to primary colors) much to Billy Goat Gruff's dismay. This week, the goats got together to discuss these awkward numbers a bit more
“I suppose,” said Billy Goat Gruff, “that since we discovered we must use these awkward numbers...”
“Primary numbers,” interrupted Awkward Goat.
“...these primary numbers,” continued Billy Goat Gruff, “it would be a good idea to have a list of them. It would make it easier to get into formation when we have a lot of goats show up for marching practice.”
“Well, I think a simple way would be to first get all the goats to line up in rows of ten.”
“Not a problem,” said Billy Goat Gruff. “Marching Society! Rows of ten! Form UP!”
“It looks like we have 60 goats today.” said Awkward Goat. “If each goat represents a counting number, then we can have the goats who are multiples of a prime sit down or something.”
“If we let some goats sit, while the others have to stand, I'll have a mutiny on my hands,” said Billy Goat Gruff. “What if I just have them turn around?”
“That's fine,” replied Awkward Goat.
“Marching Society, count off by twos!” bellowed Billy Goat Gruff. When the goats had finished, he sang out, “Number twos! Face East! (Not you, number two, you keep facing west)”
They did the same thing for three, five and seven.
“I think we're done,” said Awkward Goat.
“How so?” asked Billy Goat Gruff, “not that I'm complaining about finishing so quickly.”
“If we continued,” said Awkward Goat, “we'd eliminate multiples of eleven, right? But we've already eliminated 11 times 2, 11 times 3, 11 times 4, and 11 times 5 when we did two, three and five. And 11 times 6 is more goats than we have here.”
“That makes sense,” replied Billy Goat Gruff, “but I don't recall doing multiples of four.”
“Ah, right. We got the multiples of four when we did multiples of two. Every second multiple of two is a multiple of four,” said Awkward Goat. “In fact, we eliminated multiples of any even number when we did the twos, since that eliminated all of our even numbers.”
“So we did, so we did,” mused Billy Goat Gruff. “You're very clever, Awkward Goat.”
“Thank you,” said Awkward Goat.
“But we don't have them all,” said Billy Goat Gruff. “I want a complete list.”
“I don't think that's possible,” said Awkward Goat. “No matter how many primary numbers we listed, the list still wouldn't be complete. We could never list them all.”
“Why not?” demanded Billy Goat Gruff.
“Well, suppose we did have a complete list,” said Awkward Goat.
“Is this going to be the same kind of trick you pulled on me last week?” asked Billy Goat Gruff.
“Yes, I suppose it is,” answered Awkward Goat. “But it isn't really a trick. If we think we have a complete list, and then find a primary number that's not on the list, then it should be clear that we were wrong about having a complete list.”
“I suppose so, then,” said Billy Goat Gruff. “Proceed.”
“If we had such a complete list, we could multiply all the primary numbers together to get some large number of goats.”
"And that huge number would be a multiple of every primary number on the list, since every listed number would be one of its factors."
“And if you had exactly that number of goats, you could form them up into a neat marching formation.”
“Yes. Yes I could. I'd take some of the primary numbers and multiply them to get the number of rows, and I'd multiply the rest of the numbers to get the number of goats in each column.”
“But what happens if one more goat shows up and wants to march?”
“One more goat?”
“Yes. Now you have that huge number, plus one. There are only two possibilities for this new number. Either it is a primary number, and is not on the list, since it is bigger than any number on the list. Or, it is not a primary number, and it is the product of at least two primary numbers which are not on the list, since it is not a multiple of any of the listed numbers.”
“So in both cases...”
“We have a new primary number. Our list wasn't complete.”
Have fun in the comments.