December 11, 2008

Perfect Numbers

When playing taxman, there are numbers like 6 where the taxman will tie your first move. Another number where this happens is 28 (try it out). When a number equals the sum of its proper factors, it is called a "perfect number".

Perfect numbers are surprisingly few and far between, and mathematical knowledge about perfect numbers is also surprisingly incomplete.

The first few perfect numbers are: 6, 28, 496, 8128, and 33550336; and they quickly get big enough to stretch the limits of computers. Only a few dozen perfect numbers are known. And while mathematicians since the Greeks have suspected that there are an infinite number of perfect numbers, that conjecture has never been proven. Futhermore, all known perfect numbers are even, but it has never been established that an odd perfect number is impossible.

Despite all the mathematical intrigue, any middle schooler can do the computations to understand the pattern underlying the known perfect numbers. All you have to do is try writing the first few perfect numbers out in binary.

Following Euclid

It is easy to build intuition about all the known perfect numbers by working out a specific example.

The perfect number 496 is binary 1111000. In this form it is clear that 496 is the product of a prime made of all binary ones "1111" (a.k.a. 31 in ten-fingered society) and a power of two with the same number of digits "1000" (a.k.a. 16).

So let's add up the complete set of prime factors of 496.

Since 1111 is a prime (let's abbreviate it p), the complete list of factors of (1111*1000) is pretty short and easy to understand. First, we list the factors of 1000, and then p times each of those (still working in binary:)

1, 10, 100, 1000
p*1, p*10, p*100, p*1000

The first row sums to 1111.
The second row sums to p*1111.
And the total is (p+1)*(1111)=(1+1111)*1111=10000*1111=11110000.

With one extra zero, that sum of the factors is double our original number. But since we have included the number itself in the sum of factors, the number is half of its sum of factors. So it is perfect.

All 46 perfect numbers that have ever been found follow exactly this same pattern. Each one is the product of an all-ones prime (called a Mersenne prime) and a power of two with the same number of digits.

It is not known if there are an infinite number of Mersenne primes. And it is not known if there are any perfect numbers that do not follow this pattern. Mathematical fame awaits anybody who can find an odd perfect number.

Posted by David at December 11, 2008 10:57 PM

496 is binary 111110000.
1111000 is decimal 120.

Posted by: Matt Canning at January 3, 2010 07:18 PM
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