Conjectura
De Viquipèdia
En matemàtiques una conjectura és una afirmació matemàtica que ha estat proposada com a certa però que encara no s'ha demostrat la seva certesa o la seva falsedat. Una vegada es demostra una conjectura rep el nom de teorema.
Taula de continguts |
[edita] Conjectures famoses
Fins la seva demostració el 1995, la més famosa de totes les conjectures era el (llavors mal anomenat) últim teorema de Fermat. En el procés de demostració es demostrà també un cas del teorema de Taniyama-Shimura. Altres conjectures especialment famoses són:
- No existeixen nombres perfectes senars.
- La conjectura de Goldbach.
- La conjectura dels primers bessons.
- La conjectura de Collatz.
- La hipòtesi de Riemann.
- P ≠ NP.
- La conjectura de Poincaré.
- La conjectura abc.
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.
[edita] Counterexamples
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.
[edita] 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? 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.
[edita] 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.
[edita] Vegeu també
- Llista de conjectures

