Sunday, 1 January 2012

2012 is NOT Weird...

and you are not likely to ever see  a year number that is weird in your lifetime. Now that is weird. 

The early meaning of weird was related to turning, or falling, from the Ind-European root wer, which makes its way into lots of mathematical words; converse, inverse, diverge, etc.  That meaning seems to have quickly given way to things "befalling" an individual, and were tied to fate.  Then Shakespeare, the Matt Groening of the late 16th century, decided to use the word to describe one of his three witches, and as everybody knows, it caused another "turn" in the common usage of the word and it came to mean strange or unusual. 
So finally I can get away from literature, about which I know very little, and talk about number theory, about which I know...... Oh never mind. 

So anyway, while some folks think ALL numbers are a little weird, in number theory the term is usually applied to a subset of the abundant numbers.  Ok, a little background....

Perfect numbers are integers that are the sum of their proper divisors.  Six is the smallest since 6 = 1+2+3.  The next few, all known to the Greeks are 28, 496, and 8128.  There is a connection, known to Euclid, that ties certain prime numbers to the perfect numbers. 
OK, so not too many numbers are perfect.  What about the rest?  Well it's one of those Goldilocks things.  (College profs prefer to talk about trichotomy laws, but I'm a fan of nursery rhymes)
If the sum is too small, smaller than the integer itself, the number is called deficient.  If the sum is too large, the number is called abundant.  But if the total is "Just Right" then baby bear eats it all up because it is perfect. 
So what about weird numbers?  Well, if you take your typical abundant number, 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102 ... and more here , and look at the divisors you can almost always find a subset of the proper divisors that does add up to the original integer.
The divisors of 12, for example, are 1,2,3,4,6 and you can use 6+4+2 to get 12.  You can do that for most all of them.  But once in a while you come across a strange and unusual number (dare I say weird?).. the smallest one is 70, in which there is no subset of the divisors that will add up to the original number.  Seventy's divisors are 1, 2, 5, 7, 10, 14, and 35 go ahead, try your luck.   I'll wait...Dum de dum... tra la...tra la......  Dum de dum de dum...

Ok, Give up yet? 
So that's what a weird number is, and there really aren't many of them... I mean relatively.  Actually there may be an infinite number of them, we don't know.  How unusual? Well, between the year 70 and the year 4036,(let's see, how old will I be then...oh never mind)  there is only one more weird number (and as I told you above, it's not 2012).  If you want to look for it, look to the past, not the future.  And don't even bother with the odds.  We haven't found any odd ones yet, but then we haven't proven there are not any yet either, so if you want to tackle that problem, just look farther out in the bigger numbers...bigger than 232 at least. 
The last three years have included a prime number (always deficient) , 2011,  and an abundant composite ,2010,  as well as a deficient composite , 2012. 
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