
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 UTC6, CharlieBoo wrote: >> On Nov 16, 12:21 am, William Elliot <ma...@panix.com> wrote: >> >>> On Thu, 15 Nov 2012, CharlieBoo 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. >> >> >> >> CB > > 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?

