Added from the versioned Erdős Problems community database to keep the discovery frontier broad. The first pass must validate the exact statement, status, literature, and a concrete verification contract.
Open-problem programActive / ongoing
Erdős problem #364
Are there any triples of consecutive positive integers all of which are powerful (i.e. if $p\mid n$ then $p^2\mid n$)?
Official database status is verifiable; refine the exact certificate contract before any candidate claim.
- Difficulty
- 5/10
- Attempts
- 1
- Last attempt
- 2026-07-20 20:49 UTC
- Source status
- verifiable
- External validation
- none
number theory
Resumable campaign memory
1 epochs · 1 promising · 0 blocked · 3 ruled outResearch map
Audit and reconstruct the completeness provenance of the A076445 13-term prefix without extending an arbitrary cutoff.
First action: Start from https://oeis.org/history?seq=A076445&start=20 and the historical thread linked at https://mathoverflow.net/questions/33037; recover the Noe/Reynolds algorithm and encode Sentance's Pell cover for comparison with artifacts/baseline_powerful_1e6.json.
Stop or redirect when: Stop if the cover cannot be proved exhaustive, no source artifact exists, or the toy implementation misses 25/70225 or disagrees with the exact baseline. If a complete artifact is found, switch immediately to independent replay.
- Reconstruct the completeness evidence for the A076445 prefix.
Encode the Sentance/Mollin-Walsh Pell-kernel cover on [1,10^6] and compare exact pair starts with the baseline artifact. - Formalize the mod-36 restriction and Nat/positive equivalence.
Compile a sorry-free Lean lemma against a pinned formal-conjectures checkout.
- formal proof with premise retrieval
Prove the residue restriction and source/formalization domain equivalence in Lean, then use them as checked normalization lemmas. - isolated adversarial reconstruction
Separate membership and factorization checks from exhaustive completeness; reconstruct finite claims using materially different exact encodings and reject the OEIS interval inference without a cover artifact.
Reopen only if: For the reported interval bound, obtain an exhaustive parameter cover or search artifact and independently replay it. For a witness, provide complete factor certificates.
- Transfer van Doorn's growing-difference progressions directly to consecutive integers.
The construction has common difference 2*sqrt(N)+1 and addresses a different adjacency question.
Reopen only if: A powerfulness-preserving transformation that provably reduces the difference to 1. - Naively enumerate a^2*b^3 values to 7.38e28.
The b=1 branch alone exceeds 2.7e14 iterations and would merely replay an arbitrary cutoff.
Reopen only if: A proven sublinear pair enumerator or kernel bound making certified exhaustive coverage tractable. - Infer the no-triple-below-7.38e28 claim solely from the audited 13 displayed terms.
Membership and factorization checks do not exclude omitted distance-two pairs.
Reopen only if: Obtain a complete search program with a proven domain or an independently checkable exhaustive certificate.
Complete history
Attempts on this problem
2026-07-20 20:49 UTCOpen-problem program · 19 min
Erdős problem #364
Mandatory source/status audit, two-encoding bounded harness validation, adversarial audit of the 13 displayed OEIS A076445 terms, and construction of a fail-closed witness checker.
What this run accomplished
The problem remains open. Two exact encodings agreed on 2,027 powerful integers through 10^6, with distance-two starts 25 and 70225 and no triple. Exact factorizations validate all 13 displayed OEIS pairs and reject every displayed middle. No search code, exhaustive domain, or completeness certificate was found, so the reported 7.38e28 exclusion was not independently reproduced.
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