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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."

"Also true."

“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.

Originally posted to Orinoco on Sat Jan 28, 2012 at 07:47 AM PST.

Also republished by SciTech, Math and Statistics Geeks, J Town, and Community Spotlight.

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Comment Preferences

  •  Tips for tops (17+ / 0-)

    A little experiment of my own.

    Send great comments you find in this diary to
    Top Comments.
    Just click on The Spinning Top.
                  Top Comments posts nightly at 7pm PST.

    "The problems of incompetent, corrupt, corporatist government are incompetence, corruption and corporatism, not government." Jerome a Paris

    by Orinoco on Sat Jan 28, 2012 at 07:35:13 AM PST

    •  I wish we could put that in every diary. (6+ / 0-)

      Except that it would kill my photobucket account. (You may notice that it is the image I found and shared with the TC team.)

      And now I know why Top Comments is so popular. 7pm PST is when PSTers would be awake and alert. Around here it is 9pm ... bedtime.

      Words have meaning and your words will reflect what is in your soul.

      by JanF on Sat Jan 28, 2012 at 09:27:45 AM PST

      [ Parent ]

      •  I'm not entirely sure about alert (3+ / 0-)
        Recommended by:
        JanF, palantir, oceanview

        but still awake at 7pm, sure, mostly. I could put a copy of the image in my photobucket account, and take the hits for my diaries. I'm guessing the image source has nothing to do with its clickability.

        "The problems of incompetent, corrupt, corporatist government are incompetence, corruption and corporatism, not government." Jerome a Paris

        by Orinoco on Sat Jan 28, 2012 at 09:54:36 AM PST

        [ Parent ]

  •  never underestimate... (7+ / 0-)

    ...the goat factor.  

    ...j'ai découvert que tout le malheur des hommes vient d'une seule chose, qui est de ne savoir pas demeurer en repos dans une chambre.

    by jessical on Sat Jan 28, 2012 at 08:04:28 AM PST

  •  Heisenberg joke (12+ / 0-)

    A cop pulled over Heisenberg for speeding. The cop asked, "Do you have any idea how fast you were going?"
    Heisenberg replied, "No, but I know exactly where I am."

    Slow thinkers - keep right

    by Dave the Wave on Sat Jan 28, 2012 at 10:22:20 AM PST

  •  For those who want more of the story (2+ / 0-)
    Recommended by:
    Orinoco, 2thanks

    The diarist says

    All counting numbers . . . [are] either prime numbers or the product of prime numbers

    Although this statement is technically true, its formation is a little unfortunate and it may leave the wrong idea in some people's minds. I regret being pedantic but permit me to make things more precise.

    Let's approach things slowly. Consider the number 30, which is 2 times 3 times 5. It's the product of three prime numbers.

    Next take 15 = 3 x 5, which is the product of two prime numbers.

    Continuing in this direction, can't we say that 5 is the product of one prime number? In fact we can; a product can have a single term.

    But this is not what we were trying to convey. We're trying to make a distinction between primes and things that are definitely not primes. So we do this: We define a composite number as a product of two or more primes. So 30 and 15 are composite, but 5 is not composite. Our statement might become "every counting number is either prime or composite (and not both)".

    But this statement, alas, is wrong. The problem is the number 1. Clearly 1 is not composite. It seems reasonable to think it prime, but we can't allow that if we want something called unique factorization, which I won't go into because Orinoco may have it on the agenda. The number 1 is called a unit, neither prime nor composite. So we have: Every counting number is either prime, composite, or a unit.

    One last thing: I started by declaring the original statement to be technically correct. But again, what about 1? It's a counting number and neither a prime nor a product of primes. So what gives?

    Well, the possibly puzzling answer is that 1 is indeed a product of primes: it's a product of zero primes! An empty product has value 1, just as an empty sum has value 0. And that loophole makes the original statement true.

    •  Is this the kind of arrant pedantry (1+ / 0-)
      Recommended by:

      up with which we shall not put?

      So there is a problem with me including the word "All" before the phrase "counting numbers." Guilty as charged.

      Decades ago, I was in traffic court on some violation or other. There were three pleas available to defendants in that court. Guilty, Not Guilty, and Guilty with an Explanation. Now, there was no functional difference between Guilty and Guilty with an explanation, you still paid a fine, your car insurance still went up, but it got more people to plead guilty instead of not guilty if they had a chance to explain their extenuating circumstances to the judge.  

      Guilty with an Explanation cost the judge a bit of time up front, but saved his sitting through a bench or jury trial, if the defendant pled not guilty.

      So I plead Guilty with an Explanation:

      The goats don't march with only one goat in their formation. So when they talk about counting numbers, they are talking about counting goats in formations of 10, 25, 60 or more. They don't have a separate term for the product of primes, because they have just begun to explore prime numbers.

      Most mathematical expositions of this nature begin by setting up all the conditions and exceptions required for the general statements to be absolutely true. And mathematicians have learned to skip the boring part (exceptions and conditions) to get to the fun stuff (how does the proof work? How are the pieces fit together? Is it elegant or clunky? Can I clean up the messy bits myself?)

      Non-mathmaticians, on the other hand, start wading through the exceptions (why can't "c" be equal to zero?) and conditions (There exists an "a" and a "b" such that... huh?) that by the time they get to the swamp they've been bitten in the ass by so many alligators that they've forgotten why they are there and their main concern is to simply get somewhere the alligators aren't.

      In my view, and Awkward Goat and Billy Goat Gruff both back me up on this, the appropriate place for such folderol is in the appendix, or, as we call it here at Daily Kos, the comments.

      So, thank you for bringing it up, some people will wonder about it, and you've provided the clarity they will need. And, for bringing it up in the appropriate place, the comments section (not that you had much choice in that regard.)

      Welcome aboard, hope to see you around more. ;^)

      "The problems of incompetent, corrupt, corporatist government are incompetence, corruption and corporatism, not government." Jerome a Paris

      by Orinoco on Sat Jan 28, 2012 at 02:39:34 PM PST

      [ Parent ]

  •  I am sorry I missed 37 (2+ / 0-)
    Recommended by:
    Orinoco, oceanview

    As it is one of my favorite numbers--a prime, of course, but interesting in what it forms the basis of:
    111, 222, 333, 444, 555, 666, 777, 888 and 999 are all multiples of 37.

    Oddly, that came to me last night about 2:34.  (For whatever reason, I've been waking up lately at times made of successive numbers [the 3-digit ones, as you know, all divisible by 3.])

    Who says insominia is all that bad?

    •  That is a fascinating series of composite numbers (2+ / 0-)
      Recommended by:
      kay dub, oceanview

      which turns into an interesting pattern question: is 1111 the next number in the series?

      Is it obvious the answer is no? If so, what insight makes it obvious?

      About that clock, perhaps it gets stuck at 1:23 while you sleep, then jumps to 2:34 and gets stuck there until 3:45 and so on, so you just appear to be waking at those times. Why is your clock trying to mess with your mind?

      Anyone want to offer an explanation why three digit numbers of successive digits are always divisible by three?

      "The problems of incompetent, corrupt, corporatist government are incompetence, corruption and corporatism, not government." Jerome a Paris

      by Orinoco on Sat Jan 28, 2012 at 02:54:38 PM PST

      [ Parent ]

      •  well I first thought (1+ / 0-)
        Recommended by:

        as I always do, that I was onto something profound and difficult with the consecutive digits divisible by 3.  But then as I looked into it, it got more and more trivial.

        Turns out they don't have to be in order.  423 works as well as 234.  Or 432, or 243.

        And the trivial part can be seen algebraically:
        n + (n +1) +  (n+2) = 3n + 3, which, ah, will always be divisible by 3.

        Not earthshaking.  But still very cool to me at 5:43.  

        My clock is inhabited by Chucky.

        •  The, ah, can be proven (0+ / 0-)

          namely, that if the sum of the digits is divisible by three, then the three digit number is also divisible by three. That's one of the cool arithmetic tricks taught by rote to kids when they are learning how to divide.  

          Couple of diaries ago, we took a look at summing sequences of integers. Turns out the sum (for numbers with odd number of digits) is the middle number times the number of digits. Since the number of digits is three, the sum will be divisible by three.

          You can see that from your equation by factoring out the three

          n + (n + 1) + (n + 2) = 3n + 3 =  3(n+1)
          and, if the numbers are in sequence (n+1) is the middle number.

          Proving that the divisibility by three of the sum of digits is related to the divisibility by three of the number written with those digits is a bit more involved.  

          "The problems of incompetent, corrupt, corporatist government are incompetence, corruption and corporatism, not government." Jerome a Paris

          by Orinoco on Sat Jan 28, 2012 at 05:57:14 PM PST

          [ Parent ]

  •  Double 19 (1+ / 0-)
    Recommended by:

    If you like the lower board, slide that dart in the outer ring and you got yourself a winner!
    Yes, another darts reference!

    •  Welcome to the series (0+ / 0-)

      took me a bit to figure out your dart reference.

      "The problems of incompetent, corrupt, corporatist government are incompetence, corruption and corporatism, not government." Jerome a Paris

      by Orinoco on Sat Jan 28, 2012 at 09:58:22 PM PST

      [ Parent ]

  •  You lost me when it got to four goats. (1+ / 0-)
    Recommended by:

    I never could figure out why anyone cared about the F of X.
    But, hey, carry on.

    •  If you are not being facetious (1+ / 0-)
      Recommended by:

      I'm not sure I understand how you followed up to the point where four goats was mentioned, then lost the way on multiples of two including multiples of four.

      Multiples of two
      2,4,6,8,10,12,14,16,18, and so on
      Include multiples of four
         4,  8,    12,     16, etc.
      And multiples of other even numbers
           6,       12,         18, ...
             8,              16, ...

      Because multiples of even numbers are also even numbers.
      When we eliminated multiples of two, we eliminated all even numbers, thus eliminated multiples of all even numbers.

      "The problems of incompetent, corrupt, corporatist government are incompetence, corruption and corporatism, not government." Jerome a Paris

      by Orinoco on Sun Jan 29, 2012 at 06:49:58 AM PST

      [ Parent ]

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