Quote:
Originally Posted by Turtle
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.  |
Yeah, I recall that... vaguely. I'll have to look it over, but I don't know the particulars of why a perfect number cannot be a perfect square. It's purportedly proven here for both even and odd perfects:
http://www.goshen.edu/~dhousman/ugre...Soe%202001.doc
If this is true (and wikipedia says it is) then all perfect numbers should have an even number of factors by virtue of not being a perfect square and should likewise follow the rule that Craig proved in the last post (which I think may apply to even perfects only).
In any case, I don't see how any of this could possibly mean multiplication by unity is prohibited by number theory.
Quote:
Originally Posted by CraigD
Also, the perfect numbers are given the formula  , where  is an element of {2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, ...}, OEIS sequence A000043, the same sequence that gives the Mersenne primes,  . Therefore, the "N" in the "Nth root" of the product of the proper factors of every perfect number that Don describes will always be exactly 1 less its corresponding . For example, the product of the proper factors of the 10th perfect number,
is
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Not enough +rep to go around
~modest
