conjecture-exploration

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:

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