Conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof.
Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done.
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.
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).
Sometimes, a conjecture is called a hypothesis when it is used frequently and repeatedly as an assumption in proofs of other results.
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.
[6] The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians.
The four color theorem was ultimately proven in 1976 by Kenneth Appel and Wolfgang Haken.
Appel and Haken's approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem (i.e., if they did appear, one could make a smaller counter-example).
Appel and Haken used a special-purpose computer program to confirm that each of these maps had this property.
Showing this with hundreds of pages of hand analysis, Appel and Haken concluded that no smallest counterexample exists because any must contain, yet do not contain, one of these 1,936 maps.
In mathematics, the Weil conjectures were some highly influential proposals by André Weil (1949) on the generating functions (known as local zeta-functions) derived from counting the number of points on algebraic varieties over finite fields.
The generating function has coefficients derived from the numbers Nk of points over the (essentially unique) field with qk elements.
In mathematics, the Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.
After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv.
The proof followed on from the program of Richard S. Hamilton to use the Ricci flow to attempt to solve the problem.
Hamilton later introduced a modification of the standard Ricci flow, called Ricci flow with surgery to systematically excise singular regions as they develop, in a controlled way, but was unable to prove this method "converged" in three dimensions.
The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.
Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics.
It was essentially first mentioned in a 1956 letter written by Kurt Gödel to John von Neumann.
Gödel asked whether a certain NP-complete problem could be solved in quadratic or linear time.
