Erdős Problem #164 — The primitive-set reciprocal logarithmic sum

About 40 years old · traced to 1986

A set $A\subseteq\mathbb N$ is primitive if no member of $A$ divides another. Prove that for every primitive set $A$ whose elements are all at least $2$ ,

\sum_{a\in A}\frac{1}{a\log a}\leq \sum_{p\ \mathrm{prime}}\frac{1}{p\log p}.

Equivalent formulations 2Other wordings

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

Erdős primitive-set inequality
Among primitive sets of integers greater than one, is the weighted reciprocal sum from Erdős's conjecture maximized by the primes?
source: Open Math Problems Claimed to Be Solved with AI
Erdős conjecture for primitive sets
A subset of the integers larger than $1$ is primitive if no member divides another. Define $f(a)=1/(a\log a)$ and $f(A)=\sum_{a\in A}f(a)$ . Let $\mathcal{P}(A)$ denote the set of primes that divide some member of $A$ . Erdős conjecture. For any primitive set $A$ , we have
$f(A) \le f(\mathcal{P}(A)).$
Erdős proved that $f(A)$ is universally bounded over primitive sets, and conjectured that this bound is attained by the set of prime numbers. The conjecture remains open, although progress is known in certain cases.
source: Jared Duker Lichtman, Greg Martin and Carl Pomerance, “Primes in prime number races”, arXiv:1809.03033 (2019).

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

The conjecture that the primes give the largest weighted reciprocal sum over primitive sets has been proved, although a related technical question about the prime $2$ remains open.

The problem asks whether every primitive set $A$ satisfies $f(A)\leq f(\mathcal P)$ , where $f(A)=\sum_{a\in A}1/(a\log a)$ and $\mathcal P$ is the primes. Erdős proved uniform boundedness in $1935$ and posed the maximization question in the $1980$ s.

Known results

Erdős and Zhang, $1993$ : $f(A)<1.84$ for every primitive set.
Zhang, $1991$ : the prime-maximization statement for sets of integers with at most four prime factors counted with multiplicity.
Lichtman and Pomerance, $2019$ : $f(A)<e^\gamma=1.781\ldots$ .
Lichtman, $2023$ : the original conjecture proved.

Proofs and formalization, 2022–2026

A $2022$ arXiv paper proves $f(A)\leq f(\mathcal P)=1.6366\ldots$ , establishing the conjecture. The result is treated as settled in later work; an alternative proof by Alexeev, Barreto, Li, Lichtman, Price, Shah, Tang, and Tao appeared in $2026$ , with a related version formalized in Lean by Math Inc.

Current status (as of May 2026): The Erdős primitive-set inequality is proved; whether $2$ is Erdős strong remains open as an auxiliary question.

Sources

No solutions have been posted yet.

Erdős primitive-set inequality

Erdős conjecture for primitive sets

Primary source

Additional references

Progress summary