Mandatory source/status baseline followed by a bounded counterexample search that enumerates 3-full m=n+1, with a separate dense smallest-prime-factor oracle and reverse-orientation controls.

counterexample and witness search

Generate every 3-full m in the bounded domain exactly once from increasing prime supports and exponents at least three, factor m-1 exactly, and independently compare the generated set with a dense sieve oracle.

Hypothesis: The sparse generator is complete and duplicate-free on a nontrivial audited overlap, and either finds a target witness with n+1<=100000000 or certifies an empty hit list only on that bounded domain.

Test: Compare the complete generated 3-full set against a dense smallest-prime-factor scan through 10^6, exercise reversed examples, then enumerate all 3-full m through 10^8.

The source audit resolves the major orientation hazard, while the independent dense encoding validates the sparse generator on a meaningful overlap. The 10^8 computation is rigorous only within its explicit range and is retained as baseline infrastructure rather than novelty.

• The original 1980 question asks whether B_2(n)=n and B_3(n+1)=n+1 has no solution; n=8 belongs to the separately stated reverse question.
• The sparse and dense encodings produce the same 307 3-full integers through 10^6, with canonical set SHA-256 b1be51e2d05797f9e1e1908a378dbfd0dda61478f7399354e08d9510791f5a12.
• There is no target witness with n+1<=100000000; exactly 1,645 3-full values were exhaustively examined.
• Assuming the abc conjecture, only finitely many target pairs exist, but no effective unconditional search bound follows.

• records/attempts/2026-07-20-baseline-audit.md, SHA-256 fb5ba02cae6c4daa9cca6b42cb7e13a49a1d8027b18a0c64d76bbf7608045805
• .proof-experiments/20260720-225355-1577b9: sparse self-test through 10^5, 129 values
• .proof-experiments/20260720-225425-8b9ae9: dense/sparse equality through 10^6, 307 values, empty hit list
• .proof-experiments/20260720-225435-3a467c: exhaustive sparse search through 10^8, 1,645 values, empty hit list
• .proof-experiments/20260720-225442-7f669d: n=8 reverse fixture returns target_witness=false
• python3 -m json.tool records/research-state.json
• python3 -m py_compile tools/erdos366_discriminator.py tools/check_erdos366_discriminator.py
• git diff --check

• .proof-experiments/20260720-225355-1577b9 — self-test passed through 10^5
• .proof-experiments/20260720-225425-8b9ae9 — independent encodings agreed through 10^6
• .proof-experiments/20260720-225435-3a467c — no hit among all 1,645 3-full m<=10^8
• .proof-experiments/20260720-225442-7f669d — reverse fixture correctly rejected

tools/check_erdos366_discriminator.py uses a dense smallest-prime-factor sieve and scans every integer, materially differing from the recursive sparse generator. Through 10^6 the canonical sets and hashes agree exactly.

not_requested

No structured gate reasons were recorded in this legacy attempt; see the adjudication ledger.

Original model outcome

progress

Public classification

progress

• Walker/Pell classification of consecutive 2-full pairs -> known powerful-pair families might contain target witnesses -> the known 8 and 12167 orientations fail because their successors are not 3-full, so Pell membership alone is insufficient.
• Shiu's sparse structure of 3-full numbers -> enumerate m=n+1 rather than densely scanning n -> the sparse method examined only 1,645 candidates through 10^8 and matched a dense oracle through 10^6.
• abc radical bounds -> asymmetric exponents 1/2 and 1/3 should force conditional finiteness -> the residual exponent is (1-5 epsilon)/6>0 for epsilon<1/5, but this supplies no unconditional effective bound.

• The target orientation is n 2-full followed by n+1 3-full; n=8 is reverse-oriented.
  Erdős–Graham 1980, printed p. 68; scan SHA-256 0cbf0c32f0ab1e1c71db5121a88bac905bf976c4a6ab6bb6d7d9cf9ddd184ed3 · Exact statement and orientation · proved
• The two independent encodings agree on all 307 3-full integers through 10^6 and find no target.
  .proof-experiments/20260720-225425-8b9ae9 · All m=n+1 with 1<=m<=1000000 · computed
• No target occurs with n+1<=100000000.
  .proof-experiments/20260720-225435-3a467c; 1,645 generated values and target_hits=[] · All positive n with n+1<=100000000 · computed
• Only finitely many target pairs exist if abc is true.
  Official discussion's radical argument with epsilon<1/5 · Conditional on abc · conditional

• Use (8,9), (12167,12168), or generic Pell-generated powerful pairs as target evidence without checking orientation and exponent three.
  Known reverse examples and unfiltered consecutive-powerful families · They prove 3-full then 2-full, or only 2-full then 2-full. · .proof-experiments/20260720-225442-7f669d and the original p. 68 statement · Produce a family member whose first number independently passes 2-full and whose successor passes 3-full.
• Routine extension of the same arbitrary search cutoff as a contribution.
  Larger bounds without a recognized benchmark, structural theorem, or certificate-bearing reproduction · A finite null search cannot settle universal nonexistence and would not improve a verified best-known result. · .proof-experiments/20260720-225435-3a467c · A predeclared recognized bound, explicit expert request, or new structural pruning theorem.
• Treat OEIS A060355 alone as a certified target exclusion through 10^22.
  The current b-file without retained exhaustive-search code · It is a superset table and lacks an independently checked completeness certificate for the reported range. · https://oeis.org/A060355/b060355.txt · Recover or reproduce the enumeration with a complete generator and independent certificate.

• Analyze n+1=p^e using the factorization of p^e-1.
  Cyclotomic factor coprimality may yield a genuine obstruction or materially prune a witness branch. · Derive gcd conditions, implement a bounded certified shell, and compare with all single-prime-support cases through 10^6. · high · open
• Reconstruct the A060355 computation below 10^22.
  A certificate-bearing complete powerful-pair list could be filtered into a rigorous problem-366 exclusion. · Audit OEIS history and Donovan Johnson table provenance for code and completeness arguments. · normal · open
• Formalize a future explicit witness in Lean.
  The existing reverse example shows numeral instances of Nat.Full can be discharged by normalization. · After a witness exists, instantiate the existential and compile with no sorry or admit. · low · open

Objective: Test the single-prime-support successor branch for structural information.

First action: Write the exact factor/gcd decomposition of p^e-1 for n+1=p^e, then enumerate e>=3 and p^e<=10^6 as a regression fixture against tools/erdos366_discriminator.py.

Stop condition: Stop immediately on a verified witness; otherwise stop after the declared shell and retain the route only if it proves a local obstruction or material pruning rule.

• Investigate the structural branch n+1=p^e with prime p and e>=3 by deriving the exact cyclotomic factor and gcd conditions for p^e-1.
• Regress the structural branch against every single-prime-support case already covered through 10^6.
• Inspect the provenance of OEIS A060355's reported table below 10^22 for executable search code or a completeness argument before attempting any larger bound.

• https://www.erdosproblems.com/366 ↗
• https://mathweb.ucsd.edu/~ronspubs/80_11_number_theory.pdf ↗
• https://combinatorica.hu/~p_erdos/1976-39.pdf ↗
• https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/ErdosProblems/366.lean ↗
• https://fq.math.ca/Scanned/14-2/walker.pdf ↗
• https://doi.org/10.1017/S0017089500008351 ↗
• https://oeis.org/A060355/b060355.txt ↗
• https://arxiv.org/abs/2005.07321 ↗
• https://arxiv.org/abs/2503.21485 ↗

GPT-5.6 Sol was principal investigator. A GPT-5.6 Terra delegate supplied bounded source reconnaissance, the initial sparse script, and memo; Sol audited relied-on claims and wrote the independent dense encoding. Tools used: web search/retrieval, curl, Poppler rendering/text utilities, SHA-256, Git read-only inspection, CPython 3.12.3 exact integers, and the proof-factory experiment harness. SymPy 1.12 was inspected but was not part of the decisive canonical comparison. The project-scoped Lean 4.27.0/Formal Conjectures bootstrap installed its toolchain and cache; its final dependency build was stopped with SIGINT after about 39 minutes to respect the epoch ceiling, so no completed local Lean validation is claimed. No CAS, SAT/SMT solver, or proof assistant established a new mathematical result.; orchestration: gpt-5.6-sol principal with gpt-5.6-terra delegates.

Duration

2516.8s

Review state

not a result claim

Attempt ID

erdos-366-20260720-231157-f9a691

No human review recorded.