## Saturday, September 8, 2012

### Do the Perfect Numbers Deserve Their Title?

We have gone over numerous number sequences; Fibonaccis, primes, triangulars, squares, cubes, naturals, and probably several more. Now, it is time to add another sequence to our toolbox. This one is called the perfect numbers.

But, Fibonaccis and triangulars are already so cool. What makes the perfect numbers cooler than them? The sequence looks pretty dull anyways:

6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139950000,...

But it really is about as cool as it gets. Look at the factors of six, ignoring six itself:

1, 2, 3

What is their sum?

1 + 2 + 3 = 6

Take 28 and do the same thing:

1 + 2 + 4 + 7 + 14 = 28

1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496

1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = 8128

A perfect number is a number whose factors sum to itself. Pretty cool right? However, it isn't like the Fibonacci's right? You can't just add a number and you have the next one. Correct, but it still isn't too hard to deal with. Look at the following chart:

 Power of 2 2n -1 Product Perfect 1 1 1 2 3 6 Yes 4 7 28 Yes 8 15 120 16 31 496 Yes 32 63 2016 64 127 8128 Yes 128 255 32640 256 511 130816 512 1023 523776 1024 2047 2096128 2048 4095 8386560 4096 8191 33550336 Yes 8192 16383 134209536 16384 32767 536854528 32768 65535 2147450880 65536 131071 8589869056 Yes 131072 262143 34359607296 262144 524287 137438691328 Yes 524288 1048575 549755289600 1048576 2097151 2199022206976 2097152 4194303 8796090925056 4194304 8388607 35184367894528 8388608 16777215 140737479966720 16777216 33554431 562949936644096 33554432 67108863 2251799780130820 67108864 134217727 9007199187632130 134217728 268435455 36028796884746200 268435456 536870911 144115187807420000 536870912 1073741823 576460751766553000 1073741824 2147483647 2305843008139950000 Yes
This chart may look very confusing. Let me explain it in better detail. Basically, the first column is the powers of two. The second column is that power times two minus one. For instance 2(2) - 1 = 3.

The third column is the product of those two numbers. And if you look at all of the yeses, you will see the first several perfect numbers.

But why are they the only yeses? The answer lies in the second column. Look at the number that it is paired with.

3, 7, 31, 127, 8191, 131071, 524287, 2147483647,...

What trait do all of these numbers have? You might see it more clearly with just the first four numbers.

3, 7, 31, 127

They are all prime. In fact, the other second column numbers are composite (or one, which is neither). I don't have a proof for this, but it is one I would really like to find out. Please comment it if you know it or think you figured it out. Also, let us know if you know any perfect number identities. I would like to post some more, as this really is a fascinating property of these numbers.

1. this seems very confusing .!
cool math 4 kids

2. Why isn't 2016 considered a perfect number?
Proper Factors:1,2,4,8,16,32,63,126,252,504,1008
sum = 2016

Could it be because 63 is not prime? Yet the definition of perfect numbers simply says that the sum of the proper factors is the number itself.

What am I missing?????

1. I figured it out (I posted the message)2016 is not perfect because it is divisible by 3, 6, 9, 12, and many more multiples of three. Therefore it is an abundant number.....Cool Number.