Erdős Problem #489

Erdős problem #489 · erdosproblems.com/489

If $A$ is a forbidden-divisor set with $|A \cap [1,x]| = o(\sqrt{x})$ and $B = \{b_1 < b_2 < \cdots\}$ the sifted set, must $x^{-1} \sum_{b_i < x} (b_{i+1} - b_i)^2$ converge to a finite limit?

Result

Proved

Status

Candidate (review pending)

AI contribution

AI-discovered

Method

Argument

Field

Number Theory, Sieve Theory

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