Conjecture

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:


 * There are no odd perfect numbers
 * Goldbach's conjecture
 * The twin prime conjecture
 * The Collatz conjecture
 * The Riemann hypothesis
 * P ≠ NP
 * The Poincaré conjecture (proven by Grigori Perelman)
 * 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.

Counterexamples
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)

Mathematical journals sometimes publish the minor results of research teams having extended a given search farther than previously done before. 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 &times; 10 12 (over a million millions). In practice, however, it is extremely rare for this type of work to yield a counter-example and such efforts are generally regarded as mere displays of computing power, rather than meaningful contributions to formal mathematics.

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 was once a sceptic, and J. E. Littlewood always was). 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?

Undecidable conjectures
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.