What is Conjecture?
A '''conjecture''' is an idea in [[mathematics]] that appears ''likely'' to be true but that has not been ''proven'' to be true. After a conjecture is proven to be true, it becomes a [[theorem]]. ==Famous conjectures== Until recently, one of the most famous conjectures was [[Fermat's Last Theorem]]. The name was incorrect, because although [[Pierre de Fermat|Fermat]] said he had a proof for it, none could be found in his notes. In 1992, British mathematician [[Andrew Wiles]] found a proof for it, making it a theorem and not a conjecture. Other famous conjectures include: * There are no odd [[perfect number]]s * [[Goldbach's conjecture]] * The [[twin prime conjecture]] * The [[Collatz conjecture]] * The [[Riemann hypothesis]] * [[P versus NP]] * The [[Poincaré conjecture]] (proven by [[Grigori Perelman]]) ==Undecidable conjectures== Not every conjecture can be proven true or false. The [[continuum hypothesis]], which describes the size of certain [[infinite set]]s, is an example. It was shown to be independent of the generally accepted [[Zermelo-Fraenkel axioms|axioms of set theory]], which means that it cannot be proven true or false using those [[axiom]]s. It is therefore possible to take 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 ''does not'' 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. {{Math-stub}} [[Category:Conjectures| ]]