"Keith F. Lynch" <kfl(a)KeithLynch.net> wrote:
> The frequencies were chosen such that the sums or differences of
> any two of the eight frequencies are as far as possible from any
> of the eight frequencies. Similarly with whole-number multiples
> ("harmonics") of the eight frequencies or of their sums or
> differences. This is to prevent the phone company equipment from
> getting confused, since non-linearities in the circuits could result
> in such sum and difference frequencies appearing.
Inspired by that, I tried searching for sets of positive integers such
that the sums and differences of pairs of elements of the set weren't
equal to elements of the set, and such that integer multiples of
elements of the set weren't equal to elements of the set.
First I tried a greedy algorithm. That gave me various finite
sets, such as {2,3,7,11}, {3,4,5,11,13,19}, {4,5,6,7,17,19,27,29},
{5,6,7,8,9,19,22,23,33,34}, and {6,7,8,9,10,11,23,25,26,38,39,41}.
Eventually I found an infinite set with this property. Can anyone
else find one? I'll post my solution in a week.
Next I'll try searching for an infinite set of integers such that no
element can be generated by any arithmetic whatsoever on any of the
other elements. Not even by something like (A-B)*(A-C)/(C+C+C+B^D).
Is this a solved problem?
