At 04:22 PM 5/12/00 +0200, Pa'l La'ng wrote:
> 1) The general task (including yours) is to have the unity, i.e. the
number "1"
>as the sum of reciprocals of natural numbers.
> 2) In case the number of the summands is finite, the denominators are
divisors
>of a perfect number, - the LCM of them.- ( See: Daniel Shanks: Solved and
>unsolved
>problems in Number Theory, Vol.I.Spartan books, Washington DC.1962; page 25.)
>Easy to prove:
>the existance of such a finite summation is equivalent to the existance of
>perfect numbers.
No, this only works one way: Given a perfect number, the sum of the
reciprocals
of its divisors (excluding 1) is 1. It doesn't work the other way around;
1/2 + 1/4 + 1/6 + 1/12 = 1, but the LCM, 12, isn't perfect.
> 3) So the existance (or non-existance) of producing the unity as the sum
>of a finite number of reciprocals of odd numbers is equivalent with the
>existance (or non-existance)
>of odd perfect numbers.
If only it were that easy:
1/3 + 1/5 + 1/7 + 1/9 + 1/15 + 1/21 + 1/27 + 1/35 + 1/63 + 1/105 + 1/135 = 1,
but you can't get an odd perfect number out of it.
--
Fred W. Helenius <[EMAIL PROTECTED]>
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