Conjecture

In the metamathematics a guess is a statement of which is not clear or some time it was not clear whether it is correct or not.

Classification of the term

Any mathematical statement falls into one of the following categories:

Statements of the third class are called open questions. If the experts expect that the statement is true, it is called an open question instead of a guess. The reasons may lie in the practical use, be in numerical evidence or purely intuitive nature.

The designation of specific statements often deviates from historical reasons, such as the Bieberbach conjecture is now proved.

An important metamathematical guess is that the majority accepted formal foundation of mathematics, the ZFC set theory is consistent: If this is true, then can not prove in ZFC itself (see Gödel's incompleteness theorem ). If ZFC contain a contradiction, that there exist a statement, so that both they and their negation are provable, it follows immediately that each expressible in ZFC statement is provable (ex falso quodlibet ).

List of some conjectures

Today proven conjectures

• Even outside mathematics was famous Fermat's Last Theorem, which is probably to thank the fact that it is both easy to formulate in a way understandable to the layman as well as formal mathematical similarity with one of the most famous theorems of mathematics, the Pythagorean theorem has. After more than 300 years and numerous failed attempts, the conjecture was finally proven and has been to designate as a great Fermat's theorem. Remarkably, has always been " Fermat 's last theorem " was the English name. The proof attempts have promoted numerous developments in number theory. The road to the final proof in 1993 by Andrew Wiles and Richard Taylor led to the evidence of some much more general assumptions.
• Also easily almost bring the non- mathematician was the four-color conjecture, which quite clearly deals with the coloring of planar maps and actually has its origins in a practical question to the map coloring. It is worth noting that, historically, multiple " proofs" were presented that were detected only after several years as faulty. The first valid proof after more than 100 years, in turn, brought numerous skeptics on the plan because it was based to a large extent on the use of computers. Converted to a traditional human-readable proof of this would have been too extensive to be fully verified by an individual.
• The 1916 set up by Ludwig Bieberbach Bieberbach conjecture was proved in 1985 by Louis de Branges de Bourcia and is therefore also referred to since as a set of de Branges.
• The Burnside conjecture that all finite groups of odd order are solvable, it was proved in the 1960s by Walter Feit and John Griggs Thompson.
• The almost 100 year long open Poincaré conjecture, which had been taken yet on the list of Millennium problems in 2000, was proved in 2002 by Grigori Yakovlevich Perelman.
• The 1611 established Kepler conjecture about the densest packing of spheres may have been proven in 1998. The proof is given computers however (about 3 gigabytes of data ) in the professional world not fully recognized because of its very extensive calculations performed by a computer.

Today refuted conjectures

• The Euler's conjecture is a named after Leonhard Euler conjecture in number theory and Fermat's theorem generalizes the large. It was in 1966 refuted by a counterexample.
• Fermat number: Fermat showed that the first five Fermat numbers are primes and conjectured in 1637 that this applies to all Fermat numbers. This conjecture was disproved by Leonhard Euler in 1732. It is now believed, on the contrary, that only the first five Fermat numbers are prime. A proof of this conjecture is still pending.
• The Mertenssche conjecture states that the inequality for the sums of the partial sums of the series applies, caused by summation of the Möbius function. From the assumption the Riemann Hypothesis would follow. The conjecture was formulated by Stieltjes in 1885 in a letter to Hermite and 1985 disproved by Odlyzko and Riele.
• Two millennia has been repeatedly suggested that the parallel postulate, which had been formulated by Euclid in the 4th century BC in his elements from the other axioms and postulates its Euclidean geometry is provable. Euclid has this question - in the sense of that terminology - left open by " calling " the statement ( postulate ) and not as " inevitable and indispensable" ( Axiom ) based thought. In the 19th century it was shown that the system of axioms ( axioms and postulates ) of Euclid is consistent, but not sufficient to prove all the theorems formulated by him. Thanks to modern formulations of a geometrical axiom system " within the meaning of Euclid ", for example Hilbert's system of axioms of Euclidean geometry could be demonstrated that the parallel postulate from the other axioms and postulates of a Euclidean geometry is independent, by applying appropriate models were specified for non-Euclidean geometries.
• The conjecture of Pólya of 1919 states that there are at least as many numbers with an odd number of prime factors are up to any limit n> 1 out such as those with an even number. The smallest counterexample n = 906 150 257 was found in 1980. This illustrates that the applying a presumption of very many (even relatively large ) numbers is no guarantee of universality.

Provability unclear

• The abc- conjecture is a 1985 drawn up by Joseph Oesterlé and David Masser mathematical conjecture, which can be viewed as a generalization of Fermat's theorem is now proved big.
• The Collatz problem, also referred to as conjecture, was discovered in 1937 by Lothar Collatz. Kurtz and Simon showed in 2006 that a natural generalization of the conjecture, that is undecidable a presumption of 4th grade. Your argument is based on considerations that John Horton Conway has published 1972.
• The P- NP problem is an unsolved problem of complexity theory, it was formulated in the early 1970s by Stephen Cook and Leonid Levin. Conjecture. The assumption is (with appropriate definition of Turing machines ) proved under certain conditions by examples. It is an open question whether there are meaningful definable machines, applies.
• The Riemann Hypothesis states that all non-trivial zeros of the zeta function lie on the line. This statement could not be proven until now, however, support the supposition numerical calculations.

Provable logically independent

• The axiom of choice is usually taken to the formal foundations of mathematics. One then speaks of the Zermelo -Fraenkel set theory with the axiom of choice, ZFC shortly. Without the axiom of choice is called the set theory ZF shortly. The Axiom of Choice is independent of ZF and ZFC is consistent if and only if ZF is consistent.
• The continuum hypothesis was formulated in 1878 by Georg Cantor proved their logical independence of ZFC in the 1960s by Paul Cohen. Kurt Gödel had previously even proved the relative consistency of the Konstruierbarkeitsaxiom, which is strictly stronger than the continuum hypothesis - from ZFC can be proven when taken Konstruierbarkeitsaxioms of the continuum hypothesis, but not vice versa.
• The parallel axiom used in geometry since Euclid has long held " superfluous " for, that is, it has been suggested that it should from the other axioms can be inferred. Over 2000 years, however, all attempts to prove this conjecture. It was not until about 1826 succeeded Lobachevsky and Bolyai independently, to show the independence of the parallel axiom of Euclid's other axioms. Depending on whether one assumes the original axiom or one or another variant of its negation as an axiom, we obtain different geometries: Euclidean geometry, hyperbolic geometry and spherical geometry.

The use of conjecture as a hypothesis

A mathematical proof must only use the approved axioms, and in turn from the axioms proven records. An unproven statement may therefore be inferred. Nevertheless, there are numerous works that do just that, especially with the Riemann Hypothesis. Strictly speaking, be in such works ie statements of the form " the (eg Riemann ) If conjecture is true, then is true ... " proven.

Now, several cases can occur:

Basically it is allowed of course, already be used exclusively proven special cases. Popular solutions of the so-called Lucifer riddle often refer to the - so far unproven - Goldbach 's conjecture as a lemma, using the statement of the conjecture but only in (relatively) few long verified special cases.