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

What about 496?

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

What about 8128?

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.



3 comments:

  1. 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?????

    ReplyDelete
    Replies
    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.

      Delete