Once a conjecture has been proven, it becomes known as a theorem, and it joins the realm of known mathematical facts. Until that point in time, mathematicians must be extremely careful about their use of a conjecture within logical structures.
Until its proof in 1995, the most famous of all conjectures was the mis-named Fermat's last theorem - this conjecture only became a true theorem after its proof. In the process, a special case of the Taniyama-Shimura conjecture, itself a longstanding open problem, was proven; this conjecture has since been completely proven.
Other famous conjectures include:
- There are no odd perfect numbers
- Goldbach's conjecture
- The twin prime conjecture
- The Collatz conjecture
- The Riemann hypothesis
- P ≠ NP
- The Poincaré conjecture
- The abc conjecture
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, 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 × 1012 + 1.
Use of conjectures in conditional proofs
Sometimes a conjecture is called a hypothesis when it is used frequently and repeatedly as an assumption in proofs of other results. 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 (it is said that Atle Selberg is, and J. E. Littlewood was, sceptical). In anticipation of its eventual proof, some have proceeded to develop further proofs which are contingent on the truth of this conjecture. 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 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? In the end they must be judged as one possible problem solving technique amongst many: they amount to reducing a question to a question we have not already solved, as opposed to the standard reduction to a question we already know how to solve.
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 (or independent) 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).
In this case, if a proof uses this statement, researchers will often look for a new proof that doesn't require the hypothesis (in the same way that it is desirable that statements in Euclidean geometry be proved using only the axioms of neutral geometry, i.e. no parallel postulate.) The one major exception to this in practice is the axiom of choice -- unless studying this axiom in particular, the majority of researchers do not usually worry whether a result requires the axiom of choice.