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Topic: Reciprocals of integers summing to 1
Replies: 21   Last Post: Nov 23, 2012 8:44 PM

 Messages: [ Previous | Next ]
 gus gassmann Posts: 60 Registered: 7/26/12
Re: Reciprocals of integers summing to 1
Posted: Nov 16, 2012 9:57 AM

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?

Date Subject Author
11/16/12 Charlie-Boo
11/16/12 William Elliot
11/16/12 Charlie-Boo
11/16/12 William Elliot
11/17/12 Charlie-Boo
11/18/12 Bill Taylor
11/21/12 David Petry
11/22/12 Bill Taylor
11/22/12 Luis A. Rodriguez
11/22/12 David Petry
11/22/12 David Petry
11/23/12 Bill Taylor
11/23/12
11/23/12 Bill Taylor
11/16/12 Don Redmond
11/16/12 gus gassmann
11/16/12 billh04
11/17/12 Luis A. Rodriguez
11/17/12 Charlie-Boo
11/19/12 Luis A. Rodriguez
11/20/12 doumin
11/22/12 Bill Taylor