Perfect numbers

Factual Questions
1

A perfect number is one that is the sum of its divisors.

An interesting characteristic of perfect numbers, in decimal notation, is every known one ends in 6 or 8.

I have noticed something about these numbers in binary:

6 > 110
28 > 11100
496 > 111110000
8128 > 1111111000000
2096128 > 111111111110000000000

In short, the Euler Generator…2[1](2[2]-1) yields n number of ones followed by n-1 number of zeroes.

Anything else come up in other notations, such as heximal or octal?

  1. n-1 ↩︎

  2. n ↩︎

2

2096128 isn’t a perfect number

  1. n-1 ↩︎

  2. n ↩︎

3

Cool. Hexadecimal perfects only contain 0, 1, C, F. Octals beyond 34 only contain 0, 1, 6, 7. Perfect numbers in base infinity show no pattern.



        Octal               Decimal          Hexadecimal  
                    6                   6                6
                   34                  28               1C
                  760                 496              1F0
                17700                8128             1FC0
            177770000            33550336          1FFF000
          77777600000          8589869056        1FFFF0000
        1777777000000        137438691328       1FFFFC0000
177777777770000000000 2305843008139952128 1FFFFFFFC0000000

(To the limit of MS Excel and Calculator)

4

This is a natual consequence of a basic theorem of perfect numbers i.e. if (2^n)-1 is prime then (2^(n-1)) x ((2^n)-1) is perfect.

  1. n-1 ↩︎

  2. n ↩︎

5

Uh, sorry. I thought n=11 did generate perfect.

Anyway, Not hexidecimal…heximal: base six.

6

That’s a joke, right?

7

I’m no math wiz by any means, but I would assume that in base infinity there could never be any pattern because every number would be a unique digit.

8

I’m no whiz either, which is why I asked. But yeah, there could never be a pattern.

now that I’m sure it’s a joke:
“Base Infinity”: HA!

9

Is there any application for perfect numbers other than being notable? Or for instance, is the formula used to calculate perfect numbers used for anything in physics or mathematics, other than soley calculating perfect numbers?

10

Though that’s a true statement, it seems that “Every even perfect number’s last digit is 6 or 8” is rather a consequence of a different true statement, somewhat converse: every even perfect number is of the form (2^(n-1)) x ((2^n)-1) where (2^n) - 1 is prime.