Is it true that, for every x, if A⊆ [x,∞) is a primitive set of integers—meaning that whenever a,b∈ A and a| b, the integers a and b are associated, hence equal—then ∑_a∈ Afrac1alog a 0 and every primitive set A⊆n∈ N:x≤ n, ∑_a∈ Afrac1(alog a)<1+o(x)?

Erdős Problem #1196 — Primitive Sets Above a Threshold

At least 58 years old · documented by 1967

Is it true that, for every $x$ , if $A\subseteq [x,\infty)$ is a primitive set of integers—meaning that whenever $a,b\in A$ and $a\mid b$ , the integers $a$ and $b$ are associated, hence equal—then

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

where the $o(1)$ term tends to $0$ as $x\to\infty$ ? Formally, does there exist a function $o:\mathbb N\to\mathbb R$ with $o=o_{\mathrm{at\ top}}(1)$ such that, for every natural number $x>0$ and every primitive set $A\subseteq\{n\in\mathbb N:x\le n\}$ ,

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

References

Primary source

The Formal Conjectures Authors, Formal Conjectures (2025); independently edited from the pinned source file.

Additional references

Pinned Formal Conjectures source, Apache-2.0.

number-theory primitive-sets

InErdős Problems

Posted 2 months ago · edited 3 weeks ago

Progress summary

Refreshed 1 month ago

Claimed solved

A 2026 paper proves the conjecture, with a stronger error bound, and a Lean formalization is also reported.

The Erdős–Sárközy–Szemerédi conjecture, recorded from 1966, asks whether every primitive set above $x$ has weighted reciprocal sum at most $1+o(1)$ . The question is Erdős Problem #1196.

Known results

Lichtman (2023): upper bound $e^\gamma\pi/4+o(1)\approx1.399+o(1)$ .
Lichtman (2020): for integers with exactly $k$ prime factors, a lower bound with error $O(k^{-1/2+o(1)})$ .
Gorodetsky, Lichtman, and Wong (2024): asymptotic deficit $(c+o(1))k^22^{-k}$ , with $c\approx0.0656$ .

2026 quantitative proof

Alexeev, Barreto, Li, Lichtman, Price, Shah, Tang, and Tao state Theorem 1.1: for every primitive $A\subseteq[x,\infty)$ , $\sum_{a\in A}1/(a\log a)\leq1+O(1/\log x)$ . Their von Mangoldt-chain method proves the conjecture; a Lean formalization is reported separately. The method was suggested by GPT-5.4 Pro, while the paper presents the proof as the authors’ work.

Current status (as of May 2026): The conjecture is settled by the quantitative theorem $1+O(1/\log x)$ , with a reported Lean formalization.

GPT-5.4OpenAIsolvedevidence

Sources

No solutions have been posted yet.