---

Claim. The answer to Problem 164 is yes. For a primitive set $A$ (no member divides another, and $A\neq\{1\}$) write $f(A)=\sum_{a\in A}1/(a\log a)$, which converges by Erdős's 1935 theorem (Erdős 1935). Lichtman proves that $f(A)\leq f(\mathbb{P})=1.6366\ldots$ for every primitive $A$, where $\mathbb{P}$ is the set of primes (Theorem 1.2 of the preprint; the repository's reading is on the card Lichtman 2022). The earlier bounds were $f(A)<1.84$ by Erdős and Zhang and $f(A)<e^\gamma=1.781\ldots$ by Lichtman and Pomerance. The argument refines the Lichtman--Pomerance method through Mertens' product theorem; its new input is that a primitive set cannot contain many elements whose largest prime factor is only slightly below the element, which gains a factor $\pi/4$ on each composite element, and $e^\gamma\pi/4<f(\mathbb{P})$. The same paper proves that every odd prime $p$ is Erdős strong, meaning $f(A)\leq f(\{p\})$ whenever every element of the primitive set $A$ has least prime factor $p$, and leaves $p=2$ open; that case was later settled by the authors of the second proof on the claim page Alexeev and coauthors.

Acceptance. The site's curator, T. F. Bloom, marks the problem proved and credits this proof, which the page lists as reviewed. The paper is J. D. Lichtman, A proof of the Erdős primitive set conjecture, Forum Math. Pi 11 (2023), e18, published online 2023-06-14, a refereed journal, listed as refereed; the site's reference cites only the arXiv preprint. The site's label carries a Lean qualification; the Lean proof the formal-conjectures statement file points to formalizes the second proof, not this argument, and is linked from that claim page. The page is dated by the first version of the preprint, posted 2022-02-04.