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.