Erdős Problem #12

Erdős problem #12 · erdosproblems.com/12

Let $A \subset \mathbb{N}$ be infinite with no distinct $a, b, c \in A$ such that $a \mid (b + c)$ with $b, c > a$. Can $|A \cap [1, N]|/\sqrt{N}$ have positive lower limit? Must every such $A$ fall below $N^{1-c}$ infinitely often?

Result

Proved(see note)

Status

Partial result

AI contribution

AI-discovered

Method

Argument

Field

Extremal Number Theory

Posed by

—

Year posed

1970

Years open

56y

Solved

2026-05-21

Model

AlphaProof Nexus

Vendor

Google DeepMind

Collaborators

—

Verification

Lean-verified

Publication

Announced

Significance

10 / 100*

Disclosed cost

—

Wikipedia

No dedicated article