Let A⊂ Z_>1 be a primitive set, meaning that no member of A divides another, and let f(A):=∑_a∈ Afrac1alog a. Let P denote the set of primes. Erdős's primitive set conjecture. For any primitive set A, f(A)≤ f( P). The conjecture asks whether the primes maximize the sum f(A) among all primitive sets. It is solved in the source paper, which proves the stated inequality.

Erdős Problem #164 — Erdős primitive-set inequality

Erdős

Let $A\subset \mathbb{Z}_{>1}$ be a primitive set, meaning that no member of $A$ divides another, and let

f(A):=\sum_{a\in A}\frac{1}{a\log a}.

Let $\mathcal P$ denote the set of primes. Erdős's primitive set conjecture. For any primitive set $A$ ,

f(A)\le f(\mathcal P).

The conjecture asks whether the primes maximize the sum $f(A)$ among all primitive sets. It is solved in the source paper, which proves the stated inequality.

Equivalent formulations 2

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).

Sources & referencesView supporting material

Primary source

Jared Duker Lichtman, “A proof of the Erdős primitive set conjecture”, arXiv:2202.02384 (2024).

Additional references

4 papers in this index state this conjecture (2013–2022). The statement above is taken from the most recent of them; the others are arXiv:2007.02301, arXiv:1909.06740, arXiv:1301.0948.

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Progress summary

Refreshed 1 week ago

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.

Equivalent formulations 2

Erdős primitive-set inequality

Erdős conjecture for primitive sets

Primary source

Additional references

Progress summary