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

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.
this seems very confusing .!
ReplyDeletecool math 4 kids
Why isn't 2016 considered a perfect number?
ReplyDeleteProper 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?????
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.
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