Erdős Problem #538

Erdős problem #538 · erdosproblems.com/538

If each integer has at most $r$ representations $m = pa$ with $p$ prime and $a \in A \subseteq [1, N]$, what is the best upper bound for $\sum_{a \in A} 1/a$? The candidate proof gives the matching order $\Theta_r(\log N / \log\log N)$.

Result

Proved

Status

Candidate (review pending)

AI contribution

AI-discovered

Method

Argument

Field

Number Theory, Multiplicative Combinatorics

Posed by

—

Year posed

—

Years open

—

Solved

2026-07-13

Model

GPT-5.6 starships (Claude Fable 5 reviewer)

Vendor

OpenAI / Anthropic

Collaborators

—

Verification

Lean-verified

Publication

Announced

Significance

10 / 100*

Disclosed cost

—

Wikipedia

No dedicated article