Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 1196

claims/: The 2 claim pages of Problem 1196, one per claimant's result; the problem's standing derives from them.

Statement. Is it true that, for any $x$ , if $A\subset [x,\infty)$ is a primitive set of integers (so that no distinct elements of $A$ divide each other) then

\sum_{a\in A}\frac{1}{a\log a}< 1+o(1),

where the $o(1)$ term $\to 0$ as $x\to \infty$ ?

Status. PROVED (LEAN): the site credits a proof found by GPT-5.4 Pro at Liam Price's prompting, with the account by Alexeev, Barreto, Li, Lichtman, Price, Shah, Tang and Tao [ABLLPSTT26]; the Lean formalization by Math Inc. is reported in the thread and cited by the paper. The accepted claim is recorded on the claim page; Nat Sothanaphan's dated notes in the thread, which sharpen the constant and give a resummation proof, are the pending claim on his page.

Source. erdosproblems.com/1196, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1196, https://www.erdosproblems.com/1196.

References.

[ABLLPSTT26] B. Alexeev, K. Barreto, Y. Li, J. D. Lichtman, L. Price, J. I. Shah, Q. Tang, and T. Tao, Primitive sets and Von Mangoldt Chains: Erdős problem #1196 and beyond. arXiv:2605.00301 (2026).
[GLW24] Gorodetsky, Ofir and Lichtman, Jared Duker and Wong, Mo Dick, On Erd\H os sums of almost primes. C. R. Math. Acad. Sci. Paris (2024), 1571-1596.
[Li20] Lichtman, Jared Duker, Almost primes and the Banks-Martin conjecture. J. Number Theory (2020), 513-529.
[Li23] Lichtman, J. D., A proof of the Erdős primitive set conjecture. arXiv:2202.02384 (2023).

Formalization. Statement in formal-conjectures.

Current assessment

The question asks whether $\sum_{a\in A}1/(a\log a)\le1+o(1)$ for every primitive $A\subset[x,\infty)$ as $x\to\infty$ . The claim page records the affirmative answer in the quantitative form $1+O(1/\log x)$ : a proof produced by GPT-5.4 Pro and submitted by Liam Price in April 2026, written up as Theorem 1.1 of [ABLLPSTT26] (card, statement on its digest) and formalized in Lean by Math Inc. The standing derives from that page, accepted on the curator's credit; the paper is an arXiv preprint without a journal record, and this corpus has not built the Lean development, so neither refereed nor formalized evidence is listed. Nat Sothanaphan's three notes of 16, 20 and 21 April 2026, produced with GPT-5.4 Thinking, sharpen the bound to $1+\gamma/\log x+O(1/\log^{2}x)$ and recast the argument as a pure resummation; the paper's Remark 4.1 credits the sharper bound to him, and the notes are the pending claim on Sothanaphan's page. Przemek Chojecki's note of 15 April 2026 in the thread, Sub-Markov chain certificates for weighted antichain bounds, written with GPT-5.4, packages the bound as the $m=1$ case of its Corollary 4.4 on sets sparse on divisibility chains, taking the Mertens-type estimate from the thread's argument; it presents a framework and not a claimed resolution, so it has no claim page. Before it, Lichtman's bound $e^{\gamma}\pi/4+o(1)$ , Theorem 1.5 on his card, was the best known; the case $x=1$ is Problem 164. The lower bounds of Lichtman [Li20] and of Gorodetsky, Lichtman and Wong [GLW24] for the integers with exactly $k$ prime factors, which the site's commentary reports, show that the constant $1$ is approached and are recorded on their cards. This corpus has not reproduced or reviewed the proof, and no literature search beyond the site's page and thread is recorded.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (17)

Problem 164 The von Mangoldt chain bound for primitive sets above x Sothanaphan's sharpened constant and resummation proof problems/divisors/E1196/claims Divisibility chains as dense as the weighted sum Divisors and Multiples Divisors and Multiples divisors/gorodetsky_2024_erdos_sums_almost_primes Proposition 4.1 (p. 17): f_k = 1 + O(k/2^(k/4)) by a probabilistic argument Theorem 1.2 (p. 2): f_k = 1 - 2^(-k)(a_1 k^2 + O(k log(k+1))) with a_1 = 0.0656...divisors/lichtman_2020_almost_primes_banks_martin_conjecture Theorem 2.2 (p. 3) with its quantitative form Theorem 4.1 (p. 6): f(N_k) tends to 1 and f(N*_k) to 6/π², with error O(k^{-1/2+ε})divisors/lichtman_2022_proof_erdos_primitive_set_conjecture Theorem 1.5 (p. 3): primitive sets in [x, infinity) have f(A) at most e^gamma pi/4 + o(1)integer_sequences/alexeev_2026_primitive_sets_von_mangoldt_chains_erdos Theorem 1.1 (p. 2): a primitive set in [x, ∞) has f(A) ≤ 1 + O(1/log x)Theorem 1.3 (p. 3): odd Banks–Martin, f(A(Q)) ≤ f(N_k(Q)) for Q a set of odd primes

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