Explore a mathematical conjecture - gather evidence, find counterexamples, or build toward a proof. Triggers on "explore", "investigate", "what if", "is it true that", "find pattern", "check if"
name conjecture-exploration description Explore a mathematical conjecture - gather evidence, find counterexamples, or build toward a proof. Triggers on "explore", "investigate", "what if", "is it true that", "find pattern", "check if" 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: n LHS RHS Equal? 0 ... ... ✓/✗ 1 ... ... ✓/✗ ... ... ... ... 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: Conjecture statement (formalized) Evidence table (test cases) Patterns observed Related sequences/results (from OEIS/arXiv) Recommendation (prove / refine / abandon)