Mathematical conjecture

A mathematical statement proposed as true but not yet established by proof. A conjecture may be supported by evidence while remaining open to proof or disproof.

Mathematical proof: A deductive argument establishing that a mathematical statement follows from accepted assumptions. A proof turns a conjecture into an established theorem.
Mathematical statement: A declarative mathematical sentence that can be evaluated as true or false under an interpretation. A conjecture is a proposed truth-valued statement.
Fermat's Last Theorem: The theorem stating that no positive integers satisfy xⁿ + yⁿ = zⁿ for any integer n greater than two. It remained an open conjecture for centuries before Andrew Wiles proved it.
Inductive reasoning: Reasoning from observed cases toward a general pattern or claim. Recognizing a pattern across examples often prompts a conjecture.
Open problem: A mathematical question that has not been resolved. An unproved conjecture is an open problem until it is proved or disproved.
Counterexample: An example that shows a universal claim is false. One counterexample can disprove a conjecture asserting a property for every case.
Axiom: A foundational assumption accepted within a mathematical theory without proof inside that theory. Unlike a conjecture, an axiom is adopted as a starting assumption.
Goldbach's conjecture: The claim that every even integer greater than two is the sum of two primes. Extensive computational checks have not yet supplied a general proof.
Experimental mathematics: The use of computation and numerical exploration to discover patterns and formulate mathematical claims. Computational experiments can reveal patterns worth stating as conjectures.
Millennium Prize Problems: Seven mathematical problems announced by the Clay Mathematics Institute in 2000, each with a million-dollar prize for a solution. Several are conjectures whose resolution would settle major mathematical questions.

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