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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence Mathematical logic is a subfield of Logic and Mathematics with close connections to Computer science and Philosophical 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. In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true Until that time, mathematicians may use the conjecture on a provisional basis, but any resulting work is itself provisional until the underlying conjecture is cleared up.

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. Sir Karl Raimund Popper ( July 28 1902  &ndash September 17 1994) was an Austrian and British Philosopher and a professor In Logic and Philosophy, proposition refers to either (a the content or Meaning of a meaningful Declarative sentence A hypothesis (from Greek) consists either of a suggested explanation for a phenomenon (an event that is observable or of a reasoned proposal suggesting a possible The word theory has many distinct meanings in different fields of Knowledge, depending on their methodologies and the context of discussion. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject For the term in chemistry see Principle (chemistry. Not to be confused with Principal.

Contents 1 Famous conjectures 2 Counter Examples 3 Use of conjectures in conditional proofs 4 Undecidable conjectures 5 See also 6 External links

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. Fermat's Last Theorem is the name of the statement in Number theory that It is impossible to separate any power higher than the second into two like Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the The conjecture taunted mathematicians for over three centuries before Andrew Wiles, a Princeton University research mathematician, finally proved it in 1993, and now it may properly be called a theorem. A mathematician is a person whose primary area of study and research is the field of Mathematics. Sir Andrew John Wiles KBE FRS (born 11 April 1953 is a British Mathematician and a professor at Princeton University Princeton University is a private Coeducational research university located in Princeton, New Jersey.

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 these ideas of 'unifying conjectures' that link different subfields of mathematics, e. In mathematics a perfect number is defined as a positive integer which is the sum of its proper positive Divisors that is the sum of the positive divisors excluding Goldbach's conjecture is one of the oldest unsolved problems in Number theory and in all of Mathematics. The twin prime conjecture is a famous unsolved problem in Number theory that involves Prime numbers It states There are infinitely many primes The Collatz conjecture is an unsolved Conjecture in Mathematics. The Riemann hypothesis (also called the Riemann zeta-hypothesis) first formulated by Bernhard Riemann in 1859 is one of the most famous and important unsolved The relationship between the Complexity classes P and NP is an unsolved question in Theoretical computer science. In Mathematics, the Poincaré conjecture (French pwɛ̃kaʀe is a Theorem about the characterization of the three-dimensional sphere among Grigori Yakovlevich Perelman (Григорий Яковлевич Перельман born 13 June 1966 in Leningrad, USSR (now St The abc conjecture is a Conjecture in Number theory. It was first proposed by Joseph Oesterlé and David Masser in 1985 The Langlands program is a web of far-reaching and influential Conjectures that connect Number theory and the representation theory of certain groups There have been several attempts in history to reach a unified theory of mathematics. g. number theory and the representation theory of Lie groups; some of these conjectures have since been proved. 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 In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group

Counter Examples

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)

Mathematical journals sometimes publish the minor results of research teams having extended a given search farther than previously done before. In Mathematics, the Pólya conjecture states that 'most' (ie more than 50% of the Natural numbers less than any given number have an odd number of For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1. The Collatz conjecture is an unsolved Conjecture in Mathematics. In Mathematics, a sequence is an ordered list of objects (or events The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French 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. Computational science (or scientific computing) is the field of study concerned with constructing Mathematical models and numerical solution techniques and using computers

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. The Riemann hypothesis (also called the Riemann zeta-hypothesis) first formulated by Bernhard Riemann in 1859 is one of the most famous and important unsolved 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 In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 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). Atle Selberg ( June 14, 1917 &ndash August 6, 2007) was a Norwegian Mathematician known for his work in Analytic number John Edensor Littlewood ( 9 June 1885 &ndash 6 September 1977) was a British Mathematician, best known for his long collaboration 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. A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to

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.

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. In Mathematics, the continuum hypothesis (abbreviated CH) is a Hypothesis, advanced by Georg Cantor, about the possible sizes of Infinite This article describes cardinal numbers in mathematics For cardinals in linguistics see Names of numbers in English. Infinity (symbolically represented with ∞) comes from the Latin infinitas or "unboundedness Undecidable has more than one meaning;In Mathematical logic: A Decision problem is called (recursively undecidable if no Algorithm can Zermelo–Fraenkel set theory with the axiom of choice, commonly abbreviated ZFC, is the standard form of Axiomatic set theory and as such is the most common 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 traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive

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. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. 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. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory.

See also

• List of conjectures

External links

• All Math Words Encyclopedia grades 7-10: Conjecture
• Open Problem Garden

This is an incomplete list of mathematical Conjectures. They are divided into four sections according to their status in 2007

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