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In scientific philosophy, Karl Popper pioneered the use of the term "conjecture" to indicate a statement which is presumed to be real, true, or genuine, mostly based on inconclusive grounds, in contrast with a hypothesis (hence theory, axiom, principle), which is a testable statement based on accepted grounds.

In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. Once a conjecture is formally proven true it is elevated to the status of theorem and may be used afterwards without risk in the construction of other formal mathematical proofs. Until that time, mathematicians may use the conjecture on a provisional basis, but any resulting work is itself conjectural until the underlying conjecture is cleared up.

Famous conjectures

Until recently, the most famous conjecture was the mis-named Fermat's last theorem, mis-named because although Fermat claimed to have found a clever proof of it, none could be found among his notes after his death. The conjecture taunted mathematicians for over three centuries before Andrew Wiles at Princeton 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 '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.


Unlike the empirical sciences, formal mathematics is based on provable truth; one cannot simply try a huge number of cases and conclude that since no counter-examples could be found, therefore the statement must be true. Of course a single counter-example would immediately bring down the conjecture, after which it is sometimes referred to as a false conjecture. (c.f. Pólya conjecture)


Use of conjectures in conditional proofs


These "proofs", however, would fall apart if it turned out that the hypothesis was false, so there is considerable interest in verifying the truth or falsity of conjectures of this type. There is also something of a question mark over conditional proofs and their 'professional' status in mathematics; are they real work?

Undecidable conjectures



See also

de:Vermutung es:Conjetura fr:Conjecturehe:השערה (מתמטיקה) hu:Sejtésnl:Vermoeden pt:Conjectura ru:Гипотеза fi:Konjektuuri sv:Förmodan th:ข้อความคาดการณ์zh:猜想


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