Conjecture Exploration

Systematically investigate a mathematical conjecture before attempting a proof.

Phase 1: Understand the Statement

• • Parse the conjecture into precise mathematical form
• • Identify the domain (integers, reals, sets, etc.)
• • Identify quantifiers (∀ for all, ∃ there exists)
• • Formalize as: "For all X in DOMAIN, PROPERTY holds"

Tools to use:

• •symbolic_compute: Parse and simplify the statement
• •sequence_lookup: Check if related to known sequences
• •web_search: Check if this is a known result

Phase 2: Gather Evidence

• • Test small cases systematically

For integer conjectures, compute:

• • Test edge cases (0, 1, -1, large numbers)
• • Look for counterexamples
• • Identify patterns in the data

Tools to use:

• •symbolic_compute: Calculate values for test cases

Phase 3: Decide Direction

Based on evidence gathered:

If all cases pass + pattern is clear: → Suggest proof strategy (induction, direct, etc.) → Transition to appropriate proof skill

If counterexample found: → Report the counterexample with details → Ask: "Should we explore a weaker version?"

If pattern unclear: → Checkpoint with user → Share all findings → Ask for guidance

Phase 4: Report Findings

Present structured summary:

1. •Conjecture statement (formalized)
2. •Evidence table (test cases)
3. •Patterns observed
4. •Related sequences/results (from OEIS/arXiv)
5. •Recommendation (prove / refine / abandon)