A conjecture is a proposition that is unproven but appears correct and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis (hence theory, axiom, principle), which is a testable statement based on accepted grounds. In mathematics, a conjecture is an unproven proposition or theorem that appears correct.
Until recently, the most famous conjecture was Fermat's Last Theorem. The conjecture taunted mathematicians for over three centuries before Andrew Wiles, a Cambridge University research mathematician, finally proved it in 1995, and now it may properly be called a theorem.
Other famous conjectures include:
The Langlands program is a far-reaching web of these ideas of 'unifying conjectures' that link different subfields of mathematics, e.g. number theory and the representation theory of Lie groups; some of these conjectures have since been proved.
Formal mathematics is based on provable truth. In mathematics, any number of cases supporting a conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample would immediately bring down the conjecture. Conjectures disproven through counterexample are sometimes referred to as false conjectures (cf. Pólya conjecture).
Mathematical journals sometimes publish the minor results of research teams having extended a given search farther than previously done. For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 1012 (over a million millions). In practice, however, it is extremely rare for this type of work to yield a counterexample and such efforts are generally regarded as mere displays of computing power, rather than meaningful contributions to formal mathematics.
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