# User Contributed Dictionary

### Verb

conjectured- past of conjecture

# Extensive Definition

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
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.

## 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 Andrew Wiles, a Princeton University research mathematician, finally proved it in 1993, and now it may properly be called a theorem.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.g.
number
theory and the representation
theory of Lie groups;
some of these conjectures have since been proved.

## 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. 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.

## See also

conjectured in Simple English: Conjecture

conjectured in Danish: Formodning
(matematik)

conjectured in German: Vermutung

conjectured in Spanish: Conjetura

conjectured in French: Conjecture

conjectured in Scottish Gaelic: Baralachas

conjectured in Italian: Congettura

conjectured in Hebrew: השערה (מתמטיקה)

conjectured in Hungarian: Sejtés

conjectured in Dutch: Vermoeden

conjectured in Japanese: 予想

conjectured in Portuguese: Conjectura

conjectured in Russian: Гипотеза

conjectured in Serbian: Конјектура

conjectured in Finnish: Konjektuuri

conjectured in Swedish: Förmodan

conjectured in Thai: ข้อความคาดการณ์

conjectured in Turkish: Konjektür

conjectured in Chinese: 猜想