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Re: Reciprocals of integers summing to 1
Posted:
Nov 16, 2012 9:57 AM
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On 16/11/2012 10:10 AM, redmond@siu.edu wrote:
> On Thursday, November 15, 2012 11:39:57 PM UTC-6, Charlie-Boo wrote:
>> On Nov 16, 12:21 am, William Elliot <ma...@panix.com> wrote:
>>
>>> On Thu, 15 Nov 2012, Charlie-Boo wrote:
>>
>>>> For each n, what are the solutions in positive integers (or in
>>
>>>> integers) to (1/X1)+(1/X2) + . . . + (1/Xn)=1 ?
>>
>>>
>>
>>> x1 = x2 =..= x_n = n
>>
>>
>>
>> But there is also 1/2 + 1/3 + 1/6 = 1. I am asking for all solutions.
>>
>>
>>
>> C-B
>
> Here's another
>
> 1/2 + 1/3 + 1/7 + 1/42 = 1.
>
> don
What about 1/1 + 1/1 + 1/(-1)?
Seems to me that the original question was not very well stated.
Maybe you want x_i =/= x_j for i =/= j? That is still quite a bit more
work than I am willing to invest.
On the other hand,
M = {n | exist 0 < x_1 < x_2 < ... < x_n with sum 1/x_i = 1}
would be interesting. I offer k=2 as one positive integer that is not
contained in M. Are there others? Is there a nice characterization?
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