When a conjecture has been proven to be true, it becomes known as a theorem, and joins the realm of mathematical facts. Until that point, mathematicians must be extremely careful about using a conjecture as part of their logical structures.
For example, the Riemann hypothesis is a conjecture from number theory that (amongst other things) makes predictions about the distribution of prime numbers. Few number theorists doubt that the Riemann hypothesis is true. In anticipation of its eventual proof, some have proceeded to develop further proofs which are contingent on this conjecture being true. These are called conditional proofs: the conjectures assumed appear in the hypotheses of the theorem, for the time being.
These "proofs", however, would fall apart if it turned out that the hypothesis were false (or as noted below, undecidable); so there is considerable interest in verifying the truth or falsity of conjectures of this type.
Unlike the empirical sciences, mathematics is based on provable truth; one cannot apply the adage about "the exception that proves the rule". Although many of the most famous conjectures have been tested across an astounding range of numbers, this is no guarantee against a single counterexample, which would immediately disprove the conjecture. For example, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 10 ^{12} (over a million millions); however, it still has only the status of a conjecture - perhaps there is a counterexample awaiting researchers at 1.2 × 10^{12} + 1.
Not every conjecture ends up being proven true or false. The continuum hypothesis, which tries to ascertain the relative cardinality of certain infinite sets, was eventually shown to be undecidable from the generally accepted set of axioms of set theory. It is therefore possible to adopt this statement, or its negation, as a new axiom in a consistent manner (much as we can take Euclid's parallel postulate as either true or false).
Famous conjectures include:
The Langlands program is a far reaching web of 'unifying conjectures' that link different sub-fields of mathematics: number theory and the representation theory of Lie groups; some of these conjectures have since been proved.