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Perfect Numbers

A Perfect Number is a positive number that is equal to the sum of all positive integers that are submultiples of it, as 6, which is equal to the sum of 1, 2, and 3.

If 2ⁿ-1 is a prime number, then 2^(n-1)(2ⁿ-1) is a perfect number

Perfect Numbers

6, 1, 2, 3

28, 1, 2, 14, 4, 7

496, 1, 2, 248, 4, 124, 8, 62, 16, 31

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

33550336, 1, 2, 16775168, 4, 8387584, 8, 4193792, 16, 2096896, 32, 1048448, 64, 524224, 128, 262112, 256, 131056, 512, 65528, 1024, 32764, 2048, 16382, 4096, 8191

8,589,869,056

And the Next number has 37 digits!

Notes

Some amazing things around here

The second perfect number is summed up like this:

28 = 1³ + 3³

And the third...

496 = 1³ + 3³ + 5³ + 7³

The fourth...

8128 = 1³ + 3³ + 5³ + 7³ + 9³ + 11³ + 13³ + 15³

would the next perfect number be equal to this...

???? = 1³ + 3³ + 5³ + 7³ + 9³ + 11³ + 13³ + 15³ + 17³ + 19³ + 21³ + 23³ + 25³ + 27³ + 29³ + 31³

And then this number?

???? = 1³ + 3³ + 5³ + 7³ + 9³ + 11³ + 13³ + 15³ + 17³ + 19³ + 21³ + 23³ + 25³ + 27³ + 29³ + 31³ + 33³ + 35³ + 37³ + 39³ + 41³ + 43³ + 45³ + 47³ + 49³ + 51³ + 53³ + 55³ + 57³ + 59³ + 61³ + 63³

Perfect Numbers

Perfect Numbers

Notes

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