I think the generalization to unrestricted arithmetic is an overreach. From any integer A, you can generate any other with an expression of the form (A+A+...+A)/A. On Thu, Jan 25, 2018 at 1:08 AM, Keith F. Lynch <kfl@keithlynch.net> wrote:

"Keith F. Lynch" <kfl@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?

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