Quote:
Originally Posted by modest
Ok, my brain has had time to warm up. What Don points out is not a property of perfect numbers and it's not "one of the "deeper" reasons why these numbers are called "perfect".". If fact, every number with an even number of factors will follow Don's little rule. I pulled 30 out of my hat, but consider its factors:
(1 • 30) = 30
(2 • 15) = 30
(3 • 10) = 30
(5 • 6) = 30
Each of these multiply to the number in question. So, the result of multiplying all the factors must be some multiple of the number being factored 
A perfect number can't be a perfect square so it must have an even number of factors. Really Don?
~modest |
First & upfront, I'm not at all sure about what Don is saying which is why I'm not offering any opinions on it. Of course as we all know, I don't let little details like that stop me from saying something in response anyway.
I see the triviality of your point though Modest, and you say something further here which again rings my little bell, & that is in regard to squares & perfects. Lost amidst the many diversions in the strange numbers thread is a conjecture I posited and that Craig proved as theorem. Again I don't know if it is shedding light on the topic at hand or casting a cloud on it.
So, for what it's worth, I give you the
The Turtle-CraigD Theorem of Odd Powers of Two.
Quote:
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Originally Posted by Craig & Turtle
Quote:
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Originally Posted by Turtle
[old]Conjecture: All odd powers of 2 are either a Cube of an odd power of 2, or the sum of a Perfect-Square-multiple of a Perfect number and a Perfect Square. [(Square*Perfect)+Square]
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The conjecture can be simplified to:
All odd powers of 2 greater than 32 are at least one Perfect-Square-multiple of a Perfect number plus a Perfect Square. ... |
The rest of Craig's post and the proof is found
here.
