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Conjecture

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.

** Famous conjectures **

Until recently, the most famous conjecture was [Fermat's Last Theorem] . The conjecture taunted mathematicians for over three centuries before [Andrew Wiles] , a [Princeton University] research mathematician, finally proved it in 1995, and now it may properly be called a theorem.

Other famous conjectures include:

- The [abc conjecture]

- ['''P''' ≠ '''NP''']

- The [Collatz conjecture]

- [Beal's conjecture]

- The [Poincaré theorem] (proven by [Grigori Perelman] )

The [Langlands program] is a far-reaching web of these ideas of ' [unifying conjecture] s' that link different subfields of mathematics, e.g. [number theory] and the [representation theory] of [Lie group] s; some of these conjectures have since been proved.

** Counter Examples **

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 a * false conjectures * (cf. [Pólya conjecture] ).

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