frontiers / frontier
e1271/1271 · statement.attested · reviewer:will-blair · 2026-06-10 · null→null
Reviewable change
back to reviewErdős Problem #489 remains OPEN. Statement: Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap [1,x]\rvert=o(x^{1/2})$. Let $B=\{ n\geq 1 : a\nmid n\textrm{ for all }a\in A\}$. If $B=\{b_1 < b_2 < \cdots\}$ then is it true that $$\lim_{x \to \infty} \frac{1}{x}\sum_{b_i < x}(b_{i+1}-b_i)^2$$ exists (and is finite)? For example, when $A=\{p^2: p\textrm{ prime}\}$ then $B$ is the set of squarefree numbers, and the existence of this limit was proved by Erdős. See also [208]. Topics: number theory. Erdős prize: no. Statement is machine-verified in Lean (formal-conjectures). OEIS: N/A.
A proposal; no authority until an accepted review event.
accept gate
0 of 4 on recordtimeline
vpr_690ae23d1ab947ecSEMANTIC-EDGE DRAFT -> Erdos #488 (vf_29c8b747d0aa12af) [related, confidence 0.68]: 489's set of integers not divisible by any element of A is the exact complement of 488's set B of multiples of a finite set A, so both study the density behavior of the same multiples/non-multiples dichotomy. -- LLM-drafted (20-agent extraction, 2026-06); NOT adjudicated. Accept or reject via `vela proposals accept/reject` under reviewer authority.proposed
reason
SEMANTIC-EDGE DRAFT -> Erdos #488 (vf_29c8b747d0aa12af) [related, confidence 0.68]: 489's set of integers not divisible by any element of A is the exact complement of 488's set B of multiples of a finite set A, so both study the density behavior of the same multiples/non-multiples dichotomy. -- LLM-drafted (20-agent extraction, 2026-06); NOT adjudicated. Accept or reject via `vela proposals accept/reject` under reviewer authority.
provenance
proposed by
agent:semantic-edge-extractor
actor type
human
created at
2026-06-10
target type
finding
affected
inspect finding →Erdős Problem #489 remains OPEN. Statement: Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap [1,x]\rvert=o(x^{1/2})$. Let $B=\{ n\geq 1 : a\nmid n\textrm{ for all }a\in A\}$. If $B=\{b_1 < b_2 < \cdots\}$ then is it true that $$\lim_{x \to \infty} \frac{1}{x}\sum_{b_i < x}(b_{i+1}-b_i)^2$$ exists (and is finite)? For example, when $A=\{p^2: p\textrm{ prime}\}$ then $B$ is the set of squarefree numbers, and the existence of this limit was proved by Erdős. See also [208]. Topics: number theory. Erdős prize: no. Statement is machine-verified in Lean (formal-conjectures). OEIS: N/A.
vf_2c6d3c9f87b43ccdRead-only frontier; diff not recomputed.
Erdős problems frontier receives a reviewable source, finding, caveat, replication, evaluation, or proof-affecting edit.
The packet names affected record objects, evidence, rationale, reviewer-facing fields, and expected proof impact.
Schema, provenance, benchmark, contradiction, and proof checks decide whether the request is ready to read.
A steward accepts, rejects, caveats, revises, or retracts the request under an inspectable identity.
Only the accepted event mutates frontier state. Atlases, constellations, and search update from that record state.
Jump to a section, signal, campaign, document, primitive, work path, frontier, record index, atlas, constellation, agent, capability, or full-state search.