billh04
Posts:
7
Registered:
1/27/11


Re: Reciprocals of integers summing to 1
Posted:
Nov 16, 2012 10:07 AM


On Nov 16, 8:57 am, gus gassmann <g...@nospam.com> wrote: > On 16/11/2012 10:10 AM, redm...@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?
Since 1 = 1/2 + 1/3 + 1/6, then 1/2 = 1/4 + 1/6 + 1/12. Hence 1 = 1/2 + 1/2 + 1/4 + 1/6 + 1/12. We probably want to rule out cases like this also. More generally, 1/n = 1/2n + 1/3n + 1/6n and then 1 = 1/n + 1/n + 1/n + ... + 1/2n + 1/3n + 1/6n.

