Problem 1196

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 $x\to \infty$ ?

Source: erdosproblems.com/1196

Number theory
Primitive sets

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

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 (Price, 2026); Nat Sothanaphan's dated notes in the thread, which sharpen the constant and give a resummation proof, are the pending claim on his page.

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).

Problem 1196

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Proof claims

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