Erdős Problem #539

Erdős problem #539 · erdosproblems.com/539

For $|A| = n$, how small can the cofactor set $Q(A) = \{a / \gcd(a,b) : a, b \in A\}$ be? The answer is $h(n) = n^{1/2 + o(1)}$: a new upper bound $h(n) \le n^{1/2} \exp(O(\sqrt{\log n}))$ matches the classical lower bound.

Result

Proved(see note)

Status

Resolved

AI contribution

AI-discovered

Method

Construction

Field

Number Theory, Multiplicative Combinatorics

Posed by

—

Year posed

1973

Years open

53y

Solved

2026-06-10

Model

ProofCouncil (GPT-5.5 Pro)

Vendor

OpenAI

Collaborators

—

Verification

Lean-verified

Publication

Announced

Significance

10 / 100*

Disclosed cost

—

Wikipedia

No dedicated article