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Erdős problems frontier

constellation seal · derived from vfr_37aec80d874a0239
id
vfr_37aec80d874a0239
license
CC-BY-4.0
findings
1,256
accepted core
6
contested
0
links
17
sources
1,234
evidence
1,256
avg conf
0.98

used by 0 · replayed by 2 producers

e1271/1271 · statement.attested · reviewer:will-blair · 2026-06-10 · null→null

Reviewable change

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add a finding

verified — A frozen deterministic verifier re-checked the claim and passed.accepted

Erdős Problem #884 has been DISPROVED (a counterexample is known). Statement: For a natural number n, let $1 = d_1 < \dotsc < d_{\tau(n)} = n$ denote the divisors of $n$ in increasing order. Does it hold that $\sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i} \ll 1 + \sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i}$ for $n \to \infty`, i.e. $\sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i} \in O \left( 1 + \sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i}) \right)$? This conjecture has been **disproved**: - In September 2025, Terence Tao gave a conditional _negative_ answer assuming the prime tuples conjecture, see `erdos_884_false_of_hardy_littlewood` for this implication. - Daniel Larsen subsequently gave an [unconditional disproof](https://github.com/Larsen-Daniel/Erdos-884/blob/main/884.pdf). *Reference:* [erdosproblems.com/884](https://www.erdosproblems.com/884) Topics: number theory, divisors. Erdős prize: no. Statement is machine-verified in Lean (formal-conjectures). OEIS: N/A.

id
vpr_ceb43de800efb758
frontier
Erdős problems frontier
kind
finding.add
created
2026-05-30
findings
+1
state
null → 66650469

accept gate

1 of 4 on record
—
signature
reviewer:erdos-db-trust · no key registered on this bundle
✓
chain
null → 66650469
—
witness
no verifier attachment on record for this target
—
grade
in state · unreviewed

timeline

  1. 2026-05-30proposeproposed · finding.addagent — machine actor, no signing keyagent:erdos-spine-ingestvpr_ceb43de800efb758Candidate claim vc_a5a2f7d21f65aba4 imported from artifact packet cap_61973ee16b553d57
  2. 2026-05-30acceptfinding.assertedreviewer:erdos-db-trustreviewer:erdos-db-trustnull→66650469vev_915ba328f388440fCandidate claim vc_a5a2f7d21f65aba4 imported from artifact packet cap_61973ee16b553d57

proposed

reason

Candidate claim vc_a5a2f7d21f65aba4 imported from artifact packet cap_61973ee16b553d57

finding type

theoretical

proposed confidence

0.99

confidence basis

agent-imported candidate claim; reviewer acceptance required

provenance

proposed by

agent — machine actor, no signing keyagent:erdos-spine-ingest

actor type

agent

created at

2026-05-30

target type

finding

affected

inspect finding →

Erdős Problem #884 has been DISPROVED (a counterexample is known). Statement: For a natural number n, let $1 = d_1 < \dotsc < d_{\tau(n)} = n$ denote the divisors of $n$ in increasing order. Does it hold that $\sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i} \ll 1 + \sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i}$ for $n \to \infty`, i.e. $\sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i} \in O \left( 1 + \sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i}) \right)$? This conjecture has been **disproved**: - In September 2025, Terence Tao gave a conditional _negative_ answer assuming the prime tuples conjecture, see `erdos_884_false_of_hardy_littlewood` for this implication. - Daniel Larsen subsequently gave an [unconditional disproof](https://github.com/Larsen-Daniel/Erdos-884/blob/main/884.pdf). *Reference:* [erdosproblems.com/884](https://www.erdosproblems.com/884) Topics: number theory, divisors. Erdős prize: no. Statement is machine-verified in Lean (formal-conjectures). OEIS: N/A.

vf_a3424c338522cb4b

Diff

Read-only frontier; diff not recomputed.

Review chain

  1. 01request

    Change request

    Erdős problems frontier receives a reviewable source, finding, caveat, replication, evaluation, or proof-affecting edit.

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  2. 02packet

    Diff packet

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  3. 03checks

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  4. 04review

    Reviewer decision

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  5. 05accepted

    Accepted event

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finding.noted · reviewer:will-blair · 1 day

renders the record as of vev_d199cb2e · 1,338 events · hub

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