Conjecture - SpeedyLook encyclopedia

Anteroom of a Theorem or unstable part of the mathematical Puzzle?

When it is - after a rigorous mathematical work of demonstration - that a conjecture is true, it becomes Théorème and joined the kingdom of the made mathematical . Until this ultimate stage of veracity, the mathematicians must thus pay attention extremely when they call upon a conjecture in their logical structures and their demonstrations.

For example, the Hypothèse of Riemann is a conjecture of the Théorie of the numbers which states (inter alia things) forecasts on the distribution of the prime numbers. Few theorists of the numbers doubt owing to the fact that the assumption of Riemann is true. In waiting of its possible proof, certain mathematicians develop other demonstrations which rest on the Vérité of this conjecture. However, this “evidence” would fall of pieces if this assumption of Riemann appeared false and the fact shown could not be held for truths if the assumption appeared indécidable. There is thus a major mathematical interest to show the truth or the falseness of the hanging mathematical conjectures.

Although the majority of the most famous conjectures were checked for astonishing strings of numbers, that does not constitute a guarantee against simple a Contre-exemple, which would immediately refute the conjecture considered. For example, the Conjecture of Syracuse - which relates to the stop of a certain succession of integers - was examined for all the integers up to 1,2 × 10¹² (either more than thousand billion). However, it always has the statute of conjecture because it can always exist a counterexample of values which could be found with of 1,2 × 10¹² and which would cancel its statement.

All the conjectures do not end up being established like true or false. For example, the Hypothèse of continuous the - which tries to establish the relative cardinality some Ensemble S Infini S - proved Indécidable starting from all the Axiome S generally allowed of the Set theory. There is thus possible to adopt this Assertion, or its negation, as new axiom while remaining coherent (as we can also accept the Postulat parallel of Euclide like truth or forgery).

Famous examples of conjectures

The famous conjectures include/understand to date:

the conjecture of the four colors, which was shown in 1976 by using computer programs and completely formalized in 2006 in the Assistant of proof Coq.
the Conjecture of Kepler (see) (formulated by Johannes Kepler in 1611 and solved in 2003),
the " Last theorem of Fermat " (formulated in 1670 and solved in 1995),
the Conjecture of Goldbach (formulated in 1742),
the Assumption of Riemann (formulated in 1859),
the Conjecture of Poincaré (formulated in 1904 and solved in 2003),
the Conjecture of Syracuse (formulated in the years 1950),
the conjecture '' ABC '' (formulated in 1985),
the conjecture ''' P ''' ≠ ''' NP ''',
the Conjecture of the prime numbers twins who is the oldest unsolved conjecture.
the conjecture of the perfect numbers,
the Conjecture of Birch and Swinnerton-Dyer.

Until its proof in 1995 the, most famous of all the conjectures was that called " the Last theorem of Fermat " . It is only after its Démonstration by the mathematician Andrew Wiles that this conjecture became theorem. The demonstration consisted in proving a particular case of the conjecture of Taniyama-Shimura, problem then on standby of resolution during forty years. It was known indeed that the last theorem of Fermat rose from this particular case. The complete theorem of Taniyama-Shimura was finally shown in 1999 by Breuil, Conrad, Diamond, and Taylor which, while being based on the work of Wiles, filled by jumps of chip the remaining cases until the demonstration of the complete result.

The most discussed currently, but also oldest which can be dated, is probably the conjecture of Kepler . The proof which in summer published in the newspaper Annals off Mathematics satisfied the experts with 99%. A satisfactory proof at 100% still remains to be produced.

A conjecture which resisted during 66 years is the problem of Robbins. Its interest lies in the fact that the only solution which exists about it was produced by a computer program, (see W. McCune. Solution off the Robbins problem. J. Automated Reasoning, 19 (3): 263--276, 1997).

Examples of works in progress

The program of Langlands is a sequence of great scale which aims at the unification of the conjectures connecting various fields of mathematics: the Theory of the numbers and the theory of the representation of the groups of Dregs, some of these conjectures since having been shown.

Others

the term of conjecture should not be confused with that of Conjoncture.

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