"Fred W. Helenius" wrote:
> 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) Really, I cocentrated only to the words" weird"and " off topic" and
unforgiveably I
was unexact. The word "general" must have been replaced by the word "special".
I apologize.
> 1/2 + 1/4 + 1/6 + 1/12 = 1, but the LCM, 12, isn't perfect.
2) This is not counter-example, because multiplied by 2 we receive
1 + 1/2 +1/3+1/6 = 2 and 1/2 +1/3 +1/6 = 1 (This is the
base-equation, the LCM is perfect,
as the number "6" , too. This procedure could be repeated by any other perfect
number)
>
>
> > 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.
3) To the word "special". The unity as a sum of odd reciprocals has the
following
special necessery and sufficient conditions to be equivalent of the existance or
non-existance
of odd perfect numbers ( according to Euler):
31) The number of the summands must be odd
32) The reciprocals must be either those of squares, or those of
such other
prime-powers being not the divisors of any square(the exponents of this prime
should be
continued from the number "1" to an odd one).
> 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.
Among these denominators you find no squares, so I really can't get a perfect
number from this sum.
And to find one fulfilling Euler's conditions -- , it is not easy at all.
Best regards
Pa'l
La'ng , Budapest, Hungary
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