Erdős Problem #450

Erdős problem #450 · erdosproblems.com/450

How large must $y(\varepsilon, n)$ be so that every interval $(x, x+y)$ contains at most $\varepsilon y$ integers having a divisor in $(n, 2n)$? The candidate proof gives the sharp fixed-$\varepsilon$ order $y = \Theta_\varepsilon(n)$, uniformly in the translate.

Result

Proved

Status

Candidate (review pending)

AI contribution

AI-discovered

Method

Argument

Field

Number Theory, Divisors

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