# conjecture

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.

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.

## 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 1993, and now it may properly be called a theorem.

Other famous conjectures include: Some other Conjecture's Consist of Medians, Equidistant values, and Concurrent.

Angle Bisector Concurrency Conjecture The three angle bisectors of a triangle are concurrent.

Perpendicular Bisector Concurrency Conjecture. The three perpendicular bisectors of a triangle are concurrent.

Circumcenter Conjecture The circumcenter of a triangle is equidistant from the vertices.

Median Concurrency Conjecture The three medians of a triangle are concurrent.

Centroid Conjecture The centroid of a triangle divides each median into two parts so that the distance from the centroid to the vertex is twice the distance from the centroid to the midpoint of the opposite side.

Center of Gravity Conjecture The centroid of a triangle is the center of gravity of the triangular region.
• Circumcenter = Creates an outer circle.
• Incenter creates inside circle.
• Centroid creates the center of gravity.
The Langlands program is a far-reaching web or of there ideas 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.

## 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. PÃ³lya conjecture)

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.

## 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.

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. 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 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.

Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
proposition is the content of an assertion, that is, it is true-or-false and defined by the meaning of a particular piece of language. The proposition is independent of the of communication.
Mathematical logic is a branch of mathematics, which grew out of symbolic logic. Subfields include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic has contributed to, and been motivated by, the study of foundations of mathematics, but
theorem is a statement, often stated in natural language, that can be proved on the basis of explicitly stated or previously agreed assumptions. In logic, a theorem is a statement in a formal language that can be derived by applying rules and axioms from a deductive system.
In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. A proof is a logical argument, not an empirical one. That is, one must demonstrate that a proposition is true in all cases before it is considered a theorem of mathematics.
Philosophy of science is the study of assumptions, foundations, and implications of science. The philosophy of science may be divided into two areas: Epistemology of science and metaphysics of science.
Karl Raimund Popper, CH, FRS, FBA (July 28, 1902 – September 17, 1994) was an Austrian and British[1] philosopher and a professor at the London School of Economics.
proposition is the content of an assertion, that is, it is true-or-false and defined by the meaning of a particular piece of language. The proposition is independent of the of communication.
A hypothesis (from Greek ὑπόθεσις) consists either of a suggested explanation for a phenomenon or of a reasoned proposal suggesting a possible correlation between multiple phenomena.
The word theory has a number of distinct meanings in different fields of knowledge, depending on their methodologies and the context of discussion.

In common usage, people often use the word theory to signify a conjecture, an opinion, or a speculation.
axiom is a sentence or proposition that is not proved or demonstrated and is considered as self-evident or as an initial necessary consensus for a theory building or acceptation.

Fermat's last theorem states that:

It is impossible to separate any power higher than the second into two like powers,

or, more precisely:

Pierre de Fermat IPA: [pjɛːʁ dəfɛʁ'ma] (August 17 1601 – January 12 1665) was a French lawyer at the Parlement
Andrew Wiles

Sir Andrew John Wiles
Born March 11 1953 (age 54)
Cambridge, England
Princeton University is a private coeducational research university located in Princeton, New Jersey. It is one of eight universities that belong to the Ivy League.
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 itself.
Goldbach's conjecture is one of the oldest unsolved problems in number theory and in all of mathematics.[1] It states:

Every even integer greater than 2 can be written as the sum of two primes.

The twin prime conjecture is a famous problem in number theory that involves prime numbers. It was first proposed by Euclid around 300 B.C. and states:

There are infinitely many primes p such that p + 2 is also prime.

Collatz conjecture is an unsolved conjecture in mathematics. It is named after Lothar Collatz, who first proposed it in 1937. The conjecture is also known as the 3n + 1 conjecture, the Ulam conjecture (after Stanislaw Ulam), 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 problems in mathematics.
The relationship between the complexity classes P and NP is an unsolved question in theoretical computer science. It is considered to be the most important problem in the field--the Clay Mathematics Institute has offered a \$1 million US prize for the first correct proof.
In mathematics, the PoincarÃ© conjecture (IPA: [pwɛ̃kaˈʁe])[1] is a theorem about the characterization of the three-dimensional sphere amongst three-dimensional manifolds.
Grigori Yakovlevich Perelman
Born May 13 1966 (age 41)
Field Mathematician
abc conjecture in number theory was first proposed by Joseph OesterlÃ© and David Masser in 1985. It is stated in terms of simple properties of three integers, one of which is the sum of the other two.