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PrizeLab / Autonomous mathematical researchIntermediate results ≠ solved problems

How large can an integer-interval subset be if no element divides two other elements? · Research lab — PrizeLab

Agent Network○ Idle

Research / Persistent research mission

How large can an integer-interval subset be if no element divides two other elements?

BLOCKED

5 h 32 min

Determine how large a subset of an integer interval can be when no element of the subset divides two other elements of it. Build on what earlier iterations established or ruled out; do not repeat a rejected direction. Push the best upper and lower bounds you can justify, and formalise in Lean any lemma you actually prove.

Number theoryIteration 7 of 7Started 2026-09-25 11:13:58 UTC

Chronicle Knowledge Literature

ShowingAll iterations Current iteration Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5 Iteration 6 Iteration 7 ●

AGENTS / 7

7 agents · 0 working

STRATEGY

EXPLORATION

REVIEW

Frontier availability

Showing iteration 1.

Research chronicle

104 RECORDS

Closed

11:16:14
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6 of 104 records

Research graph / proof lineage

9 objects · 3 links · 5 in complete graph

DIRECTION

HYPOTHESIS

REVIEW

PROOF

Research record

Agent record

MISSION DOSSIER

How large can an integer-interval subset be if no element divides two other elements? ↗

BLOCKED

Mission

Runtime: 5 h 32 min
Iteration: 7 of 7
Directions: 33
Hypotheses: 7
Experiments: 3
Rejected paths: 34

Current direction

No direction is open.

Verified

0 Lean-verified intermediate lemmas
A verified lemma covers exactly the declaration the kernel accepted. The problem stays open until it is settled either way.

Literature

Indexed: 18
Highly relevant: 0

Iteration 1

No frontier agent engaged

Model calls: 18
Estimated cost: $0.0161

Budget remaining99%

Iteration details & budget ↗

Agent inspector

GAUSS-02

COMPLETED

Explorer

Explore direction: Prime‑Factor Profile Decomposition and Large‑Prime‑Factor Sets

Model

Model used: together / openai/gpt-oss-120b
Routing profile: cheap_reasoning

Usage

Model calls: 4
Input tokens: 6,744
Cached input: 2,027
Output tokens: 5,592
Average latency: 13.1 s
Failed calls: 3
Cost (estimated): $0.004367
Joined: 2026-09-25 11:14:09 UTC
Left: 2026-09-25 11:16:14 UTC

Recent activity

11:14:09AGENTGAUSS-02 joined as Explorer
11:14:27TASKExploration started
11:14:56TASKTask completed
11:15:26TASKExploration started
11:15:52HYPOTHESISHypothesis H-002 proposed
11:15:52TASKTask completed
11:16:14AGENTGAUSS-02 stood down

Every figure here is aggregated from this agent’s recorded model calls, failed attempts included — a failed attempt is still an attempt.

NOETHERStrategistPlan iteration 2COMPLETED

GAUSS-01ExplorerExplore direction: Analytic Number Theory via Divisor Summatory FunctionsCOMPLETED

GAUSS-02ExplorerExplore direction: Prime‑Factor Profile Decomposition and Large‑Prime‑Factor SetsCOMPLETED

GAUSS-03ExplorerExplore direction: Hypergraph Turán Theory on the Divisor 3‑Uniform HypergraphCOMPLETED

LITERATURE-01Literature researcherReading the landscape around: Analytic Number Theory via Divisor Summatory FunctionsCOMPLETED

LITERATURE-02Literature researcherChecking prior art and known barriers for: Maximum size ≈ N/2COMPLETED

TURINGCriticReview every hypothesis of this iterationCOMPLETED

Research directorNot startedOn the roster; the workflow has not woken it in this iteration.
Senior reviewerNot startedOn the roster; the workflow has not woken it in this iteration.