Andy Latto <andy.latto@pobox.com> wrote:

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Eventually I found an infinite set with this property. Can anyone else find one? I'll post my solution in a week. . . . . . . . . . . . . . . . . . . . . . . The odd primes work.

I missed that solution, but I came close. First I found a solution for just the sums and differences, but not the multiples: Numbers congruent to 1 mod 3: {1,4,7,10,13,16,19,...} Their sums and differences are all congruent to 2 mod 3, so their sums and differences aren't in the set. Then I realized if I restricted it to just the primes in that set, {7,13,19,31,...}, I'd also get the multiples, since no prime is the multiple of another prime. I could prepend 2 and 3 to this set, since the differences between elements (other than the prepended 2 and 3) are always multiples of 6. So {2,3,7,13,19,31,...}. By a theorem by the brother in law of the composer of A Midsummer Night's Dream and Fingal's Cave, that set is infinite. Then I realized that the primes congruent to 2 mod 3, {2,5,11,17,23,...}, would also work. But I completely missed that the set of all odd primes would work. Yours is the best solution because it has the most elements less than N for all large N. Can you find a solution for a dense set of positive real numbers? Odd primes divided by odd primes works for the sums and differences, since the sum (and difference) of two odd-over-odd fractions is always an even-over-odd fraction, and no odd-over-odd fraction can equal any even-over-odd fraction. But what about multiples? An even-over-odd fraction can be an even multiple of an odd-over-odd fraction. For instance 5/3 + 17/3 = 2 * (11/3), so that doesn't work. For extra credit, find a solution for a dense set of positive and negative real numbers. Note that if X is in the set -X can't be in the set, since their sum would be zero, which is of course an integer multiple of every element.