Second-order asymptotics in the corrected Erdős Problem 796
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Let g₃(n) be the largest size of A⊂{1,…,n} such that each integer m has fewer than three representations m=a₁a₂ with distinct a₁<a₂ in A. Beyond g₃(n)∼n log log n/log n, does the residual divided by n/log n converge to a constant? This uses the corrected log n scale, not the printed log-squared scale.
Why it matters and what counts as progress
Why it matters: The second-order coefficient refines a classical multiplicative-representation extremal asymptotic; the corrected scale is essential to the question.
What counts as progress: Prove convergence of the corrected second-order residual, determine its constant or improve matching error bounds.
How the problem developed
- 1964 · Partial result [1]
Paul Erdős Established the leading asymptotic g₃(n)∼n log log n/log n for the largest subset A of {1,…,n} in which each integer has fewer than three factorizations m=a₁a₂ with a₁<a₂ in A.
- 1969 · Open question [1]
Paul Erdős Asked for a second-order term; the printed problem has a log-squared denominator, while the corrected formulation uses log n.
- 18 Oct 2026 · Proof claim [2]
Star Fleet Reported convergence of the normalized second-order residual to an explicit constant written as Mertens’ constant plus a variational limit.
AI contribution
Star Fleet reports Erdős Problem 796 second-order constant
For the corrected Erdős Problem 796 formulation, Star Fleet reports that (g₃(n)−n log log n/log n)/(n/log n) converges to an explicit constant written Mertens.M plus a variational limit.
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