conjecture

A conjecture is a proposition that is believed to be true based on observations or heuristic reasoning but has not yet been proven. In mathematics a conjecture is a precise statement that remains unproven and may be true or false; it can be refuted by a single counterexample or supported by accumulating evidence until a proof is found. Conjectures often arise from patterns noticed in data, numerical experiments, or theoretical insight, and they frequently guide further investigation.

In mathematical practice, a conjecture sits between an observation and a theorem. A hypothesis is a tentative explanation tested by experiments, while a theorem is a statement that has been proven from axioms. A conjecture may become a theorem if a valid proof is discovered, or it may be disproven and become a counterexample that closes that particular question. Well-known conjectures that became theorems include Fermat’s Last Theorem (proven in 1994/1995) and the Poincaré conjecture (proven in 2003). Other famous conjectures, such as Goldbach’s conjecture (1742) and the Riemann Hypothesis (1859), remain unproven to this day.

In science and broader discourse, conjecture denotes an educated guess or provisional explanation subject to testing and refinement. While the term occasionally overlaps with hypothesis, conjectures in mathematics emphasize a precise, testable statement awaiting proof or disproof, whereas in empirical science they frame provisional ideas to be validated by evidence.

Etymology-wise, conjecture comes from Latin coniectura, meaning a guess or inference, via Old French into English, where the term has been used since early modern mathematics to denote unsolved but plausible statements.