In mathematics, a conjecture is a mathematical statement which appears likely to be true, but has not been formally proven to be true under the rules of mathematical logic. Once a conjecture is formally proven true it is elevated to the status of theorem and may be used afterwards without risk in the construction of other formal mathematical proofs. Until that time, mathematicians may use the conjecture on a provisional basis, but any resulting work is itself conjectural until the underlying conjecture is cleared up. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... This article documents the word proposition as it is used in logic, philosophy, and linguistics. ... Mathematical logic is a subfield of mathematics that is concerned with formal systems in relation to the way that they encode intuitive concepts of mathematical objects such as sets and numbers, proofs, and computation. ... Look up theorem in Wiktionary, the free dictionary. ... In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ...

In scientific philosophy, Karl Popper pioneered the use of the term "conjecture" to indicate a proposition which is presumed to be real, true, or genuine, mostly based on inconclusive grounds, in contrast with a hypothesis (hence theory, axiom, principle), which is a testable statement based on accepted grounds. Philosophy of science is the study of assumptions, foundations, and implications of science, especially in the natural sciences and social sciences. ... Sir Karl Raimund Popper, CH, FRS, FBA, (July 28, 1902 – September 17, 1994), was an Austrian born naturalized British[1] philosopher and a professor at the London School of Economics. ... This article is about the word proposition as it is used in logic, philosophy, and linguistics. ... Look up Hypothesis in Wiktionary, the free dictionary. ... The word theory has a number of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion. ... An axiom is a sentence or proposition that is not proved or demonstrated and is considered as obvious or as an initial necessary consensus for a theory building or acceptation. ... A principle (not principal) is something, usually a rule or norm, that is part of the basis for something else. ...

Famous conjectures

Until recently, the most famous conjecture was the mis-named Fermat's last theorem, mis-named because although Fermat claimed to have found a clever proof of it, none could be found among his notes after his death. The conjecture taunted mathematicians for over three centuries before a British mathematician Andrew Wiles working at Princeton finally proved it in 1995, and now it may properly be called a theorem. Pierre de Fermats conjecture written in the margin of his copy of Arithmetica proved to be one of the most intriguing and enigmatic mathematical problems ever devised. ... Pierre de Fermat Pierre de Fermat IPA: (August 17, 1601–January 12, 1665) was a French lawyer at the Parlement of Toulouse, France, and a mathematician who is given credit for early developments that led to modern calculus. ... For the French mathematician with work in the area of elliptic curves, see André Weil. ... Princeton University is a private coeducational research university located in Princeton, New Jersey, in the United States of America. ...

Other famous conjectures include:

• There are no odd perfect numbers
• Goldbach's conjecture
• The twin prime conjecture
• The Collatz conjecture
• The Riemann hypothesis
• P ≠ NP
• The Poincaré conjecture (proven by Grigori Perelman)
• The abc conjecture

The Langlands program is a far-reaching web of 'unifying conjectures' that link different subfields of mathematics, e.g. number theory and the representation theory of Lie groups; some of these conjectures have since been proved. In mathematics, a perfect number is defined as an integer which is the sum of its proper positive divisors, that is, the sum of the positive divisors not including the number. ... Goldbachs conjecture is one of the oldest unsolved problems in number theory and in all of mathematics. ... The twin prime conjecture is a famous problem in number theory that involves prime numbers. ... The Collatz conjecture is an unsolved conjecture in mathematics. ... Unsolved problems in mathematics: Is the real part of a non-trivial zero of the Riemann zeta function always ½? In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. ... Diagram of complexity classes provided that P ≠ NP. The existence of problems outside both P and NP-complete in this case was established by Ladner. ... In mathematics, the Poincaré conjecture (IPA: [])[1] is a conjecture about the characterization of the three-dimensional sphere amongst three-dimensional manifolds. ... Grigori Yakovlevich Perelman (Russian: ), born 13 June 1966 in Leningrad, USSR (now St. ... Unsolved problems in mathematics: For every ε > 0, does there exist a K>0 such that for every triple of coprime positive integers a+b=c, with product d of their distinct prime factors, |a|+|b|+|c| < Kd1+ε? The abc conjecture in number theory was first proposed by Joseph Oesterlé and... In mathematics, the Langlands program is a web of far-reaching and influential conjectures that connect number theory and the representation theory of certain groups. ... In mathematics, there have been many attempts down the centuries to unify the whole subject. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ...

Counterexamples

Unlike the empirical sciences, formal mathematics is based on provable truth; one cannot simply try a huge number of cases and conclude that since no counter-examples could be found, therefore the statement must be true. Of course a single counter-example would immediately bring down the conjecture, after which it is sometimes referred to as a false conjecture. (c.f. Pólya conjecture) The Pólya Conjecture is an resolved and disproved conjecture in mathematics named after the Hungarian mathematician George Pólya, who stated it in 1919. ...

Mathematical journals sometimes publish the minor results of research teams having extended a given search farther than previously done before. For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 10 ¹² (over a million millions). In practice, however, it is extremely rare for this type of work to yield a counter-example and such efforts are generally regarded as mere displays of computing power, rather than meaningful contributions to formal mathematics. The Collatz conjecture is an unsolved conjecture in mathematics. ... In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ... The integers are commonly denoted by the above symbol. ... Computational Science is the use of computers to perform research in other fields. ...

Use of conjectures in conditional proofs

Sometimes a conjecture is called a hypothesis when it is used frequently and repeatedly as an assumption in proofs of other results. For example, the Riemann hypothesis is a conjecture from number theory that (amongst other things) makes predictions about the distribution of prime numbers. Few number theorists doubt that the Riemann hypothesis is true (it is said that Atle Selberg was once a sceptic, and J. E. Littlewood always was). In anticipation of its eventual proof, some have proceeded to develop further proofs which are contingent on the truth of this conjecture. These are called conditional proofs: the conjectures assumed appear in the hypotheses of the theorem, for the time being. Unsolved problems in mathematics: Is the real part of a non-trivial zero of the Riemann zeta function always ½? In mathematics, the Riemann hypothesis (also called the Riemann zeta-hypothesis), first formulated by Bernhard Riemann in 1859, is one of the most famous unsolved problems. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... Atle Selberg (born June 17, 1917) is a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory. ... John Edensor Littlewood (June 9, 1885 - September 6, 1977) was a British mathematician. ... Conditional proof is a proof that takes the form of asserting a conditional, and proving that the premise or antecedent of the conditional necessarily leads to the conclusion. ...

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. There is also something of a question mark over conditional proofs and their 'professional' status in mathematics; are they real work?

Undecidable conjectures

Not every conjecture ends up being proven true or false. The continuum hypothesis, which tries to ascertain the relative cardinality of certain infinite sets, was eventually shown to be undecidable (or independent) from the generally accepted set of 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 we can take Euclid's parallel postulate as either true or false). In mathematics, the continuum hypothesis is a hypothesis about the possible sizes of infinite sets. ... Aleph-0, the smallest infinite cardinal In mathematics, cardinal numbers, or cardinals for short, are a generalized kind of number used to denote the size of a set. ... The infinity symbol ∞ in several typefaces. ... In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... Undecidable has more than one meaning: In mathematical logic: A decision problem is undecidable if there is no known algorithm that decides it. ... The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ... An axiom is a sentence or proposition that is not proved or demonstrated and is considered as obvious or as an initial necessary consensus for a theory building or acceptation. ... Euclid (Greek: ), also known as Euclid of Alexandria, was a Greek mathematician of the Hellenistic period who flourished in Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323 BC-283 BC). ... a and b are parallel, the transversal t produces congruent angles. ...

In this case, if a proof uses this statement, researchers will often look for a new proof that doesn't 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. Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician [[Euclid]] of Alexandria. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...

See also

• List of conjectures

This is a list of mathematical conjectures, by Wikipedia page. ...

External links

• All Math Words Encyclopedia: Conjecture