Erdős Problem #796

Erdős Problem

www.erdosproblems.com/796

The problem

Let $k\geq 2$ and let $g_k(n)$ be the largest possible size of $A\subseteq \{1,\ldots,n\}$ such that every $m$ has $<k$ solutions to $m=a_1a_2$ with $a_1<a_2\in A$ .Is it true that $g_3(n)=\frac{\log\log n}{\log n}n+(c+o(1))\frac{n}{\log n}$ for some constant $c$ ?

(Erdős Problem #796 — number theory — https://www.erdosproblems.com/796)

Result

(g₃(n) − n·loglog n/log n)/(n/log n) → M + variationalLimit — the second-order term of g₃ converges to an explicit constant (answers the open follow-up on the problem thread)

Report

Erdős Problem #796 — exact second-order constant (accepted 2026-07-14)

Proved an affirmative answer to Erdős Problem 796 in Lean. The closed theorem Erdos796.erdos796_statement : Erdos796.Statement proves that the faithful normalized residual converges to Mertens.M + variationalLimit, hence the requested constant exists. The two-gate upper reduction is closed by smoothRemainderGate_proved and the new extractedTailGate_proved; F-032 supplies the matching lower bound.

Exact verifier command: cd /root/snapshot && check_answer/verify.sh workspace/experiments/experiment_4_upper_decomposition/lean Most recent output ends: Build completed successfully (8737 jobs). PASS: authored sources are placeholder-free and Lean accepts the project

Axiom audit for both Erdos796.extractedTailGate_proved and Erdos796.erdos796_statement reports exactly [propext, Classical.choice, Quot.sound], with no sorryAx. Final verified record: F-054.

Referee

The agent claims a Lean-verified affirmative answer to Erdős 796: a closed theorem erdos796_statement proving ∃c with (g₃(n) − n·loglog n/log n)/(n/log n) → c (c = Mertens.M + variationalLimit). I audited the formal statement line-by-line against problem.md (faithful, quantifier-for-quantifier, identical to the pre-registered canonical file), re-ran the verifier myself (PASS, 8737 jobs), independently re-ran #print axioms (only propext/Classical.choice/Quot.sound — no sorryAx and no ofReduceBool, so the native_decide lemmas and the sorried PrimeNumberTheoremAnd declarations are provably outside the final proof's dependency cone), and re-elaborated Basic.lean and CanonicalTail.lean live from source. The proof is closed, hypothesis-free, machine-checked, and answers exactly the posed problem.

Independent verification (separate hardware)

Full environment reconstruction on a fresh GCP box (project + pinned sorry-free PrimeNumberTheoremAnd fork): lake build clean, pinned verifier PASS, #print axioms Erdos796.erdos796_statement → [propext, Classical.choice, Quot.sound]. The workspace's incidental native_decide lemmas and upstream sorried PNT declarations are provably outside the dependency cone.

Public bundles

https://pub-23f3a588e04a481196ed22d7e3a6f48d.r2.dev/verify/erdos-796/erdos-796-solution.zip (+ verify-cursor / verify-claude-code / verify-codex)

Download Full Solution & Verify with Your AI

The download contains everything needed to verify this solution independently: the original problem, the formal statement, the complete pinned Lean project, and the verifier script — plus step-by-step instructions for your agent. About 20 minutes, mostly downloading Mathlib.

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