Tuesday, September 25, 2007

Not perfect: an abundance of deficiency

Perfect numbers are in the news.

- The NFL is starting to think that three is the perfect number for pre-season games.

- While attempting to shoot a 59 at the PGA Tour Championship, Zach Johnson said he had the perfect club and the perfect number on the 18th hole. But his stroke wasn’t perfect and Johnson had to settle for 60.

- A post on the New York Times asks, “Is it possible that right now we have exactly the perfect number of troops in Iraq, neither more nor fewer than we need?”

A perfect number is actually an integer that is the sum of its proper divisors. The smallest perfect number is 6: which is the sum of 1, 2, and 3. The next is 28, then 496. Only 44 perfect numbers are known. Of those, they’re all even and the largest has over 19 million digits. How do those mathematicians figure that out?

Usually a number is different than the sum of its proper divisors. When the sum is greater, the number is called abundant. For example, 12 is abundant because the sum of its proper divisors 1, 2, 3, 4, and 6 is 16. When the sum is less than the number, the number is deficient. Of the first 17 integers all are deficient, except for 6, which is perfect, and 12, which is abundant.

Which leads us to believe that there may be an abundance of deficient numbers. Well, nothing is perfect, not even the NFL, the PGA, or the New York Times.

No comments: