Editing Conjecture (section)

==Resolution of conjectures==

===Proof===
Formal mathematics is based on ''provable'' truth. In mathematics, any number of cases supporting a [[universally quantified]] conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single [[counterexample]] could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the [[Collatz conjecture]], which concerns whether or not certain [[sequence]]s of [[integer]]s terminate, has been tested for all integers up to 1.2 &times; 10<sup>12</sup> (1.2 trillion). However, the failure to find a counterexample after extensive search does not constitute a proof that the conjecture is true—because the conjecture might be false but with a very large minimal counterexample.

Nevertheless, mathematicians often regard a conjecture as strongly supported by evidence even though not yet proved. That evidence may be of various kinds, such as verification of consequences of it or strong interconnections with known results.<ref>{{cite journal |last1=Franklin |first1=James  |date=2016 |title=Logical probability and the strength of mathematical conjectures |url=https://web.maths.unsw.edu.au/~jim/logicalprobabilitymathintelldraft.pdf |archive-url=https://web.archive.org/web/20170309031840/http://web.maths.unsw.edu.au/~jim/logicalprobabilitymathintelldraft.pdf |archive-date=2017-03-09 |url-status=live |journal=Mathematical Intelligencer |volume=38 |issue=3 |pages=14–19 |doi=10.1007/s00283-015-9612-3 |s2cid=30291085 |access-date=30 June 2021}}</ref>

A conjecture is considered proven only when it has been shown that it is logically impossible for it to be false. There are various methods of doing so; see [[Mathematical proof#Methods of proof|methods of mathematical proof]] for more details.

One method of proof, applicable when there are only a finite number of cases that could lead to counterexamples, is known as "[[Proof by exhaustion|brute force]]": in this approach, all possible cases are considered and shown not to give counterexamples. In some occasions, the number of cases is quite large, in which case a brute-force proof may require as a practical matter the use of a computer algorithm to check all the cases. For example, the validity of the 1976 and 1997 brute-force proofs of the [[four color theorem]] by computer was initially doubted, but was eventually confirmed in 2005 by [[theorem-proving]] software.

When a conjecture has been [[mathematical proof|proven]], it is no longer a conjecture but a [[theorem]]. Many important theorems were once conjectures, such as the [[Geometrization conjecture|Geometrization theorem]] (which resolved the [[Poincaré conjecture]]), [[Fermat's Last Theorem]], and others.

===Disproof===
Conjectures disproven through counterexample are sometimes referred to as ''false conjectures'' (cf. the [[Pólya conjecture]] and [[Euler's sum of powers conjecture]]). In the case of the latter, the first counterexample found for the n=4 case involved numbers in the millions, although it has been subsequently found that the minimal counterexample is actually smaller.

===Independent conjectures===
Not every conjecture ends up being proven true or false. The [[continuum hypothesis]], which tries to ascertain the relative [[cardinal number|cardinality]] of certain [[infinite set]]s, was eventually shown to be [[Independence (mathematical logic)|independent]] from the generally accepted set of [[Zermelo–Fraenkel 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 [[Euclid]]'s [[parallel postulate]] can be taken either as true or false in an axiomatic system for geometry).

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. without the parallel postulate). The one major exception to this in practice is the [[axiom of choice]], as the majority of researchers usually do not worry whether a result requires it—unless they are studying this axiom in particular.